objective type

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Questions:

Multiple Choice Question

94 x 1 = 94

The locus of a point which is collinear with the points (a, 0) and (0, b) is ______________

(a)

x + y = 1

(b)

\(\frac{x}{a}+\frac{y}{b}=1\)

(c)

x + y = ab

(d)

\(\frac{x}{a}-\frac{y}{b}=1\)

If the co-ordinates of a variable point p be \((t+\frac{1}{t},t-\frac{1}{t})\) where t is the parameter then the locus of p ______________

(a)

xy = 1

(b)

x²+ y²= 4

(c)

x²- y²= 4

(d)

x²- y²= 8

The locus of a point which is equidistant from (1, 0) and (-1, 0) is ______________

(a)

x-axis

(b)

y-axis

(c)

y = x

(d)

y = -x

The locus of a point which is equidistant from (-1, 1) and (4, 2) is ______________

(a)

5x + 3y + 9 = 0

(b)

5x + 3y - 9 = 0

(c)

3x - 5y = 0

(d)

3x + 5y - 9 = 0

If O is the origin and Q is a variable point on y²= x, then the locus of the mid-point of OQ is ______________

(a)

y²= 2x

(b)

2y²= x

(c)

4y²= x

(d)

y = 2x²

If p is the length of perpendicular from origin to the line \(\frac{x}{a}+\frac{y}{b}=1\) then ______________

(a)

\(\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}\)

(b)

\(\frac{1}{p^2}=\frac{1}{a^2}-\frac{1}{b^2}\)

(c)

\(\frac{1}{p^2}=-\frac{1}{a^2}+\frac{1}{b^2}\)

(d)

\(\frac{1}{p^2}=-\frac{1}{a^2}-\frac{1}{b^2}\)

The co-ordinates of a point on x + y + 3 = 0 whose distance from x + 2y + 2 = 0 is \(\sqrt 5\), is ______________

(a)

(9, 6)

(b)

(-9, 6)

(c)

(6, -9)

(d)

(-9, -6)

The point (2, 1) and (-3, 5) are on ______________

(a)

Same side of the line 3x - 2y + 1 = 0

(b)

Opposite sides of the line 3x - 2y + 1 = 0

(c)

On the line 3x - 2y + 1 = 0

(d)

On the line x + y = 3

If one of the lines by 6x²- xy + 4cy²= 0 is 3x + 4y = 0, then c = ______________

(a)

(b)

-1

(c)

(d)

-3

10.

If one of the lines of my²+(1-m²)xy - mx²= 0 is a bisector of the angle between the lines xy = 0 then m is ______________

(a)

\(\frac{-1}{2}\)

(b)

-2

(c)

(d)

11.

The distance between the parallel lines 3x - 4y + 9 = 0 and 6x - 8y -15 = 0 is ______________

(a)

\(\frac{-33}{10}\)

(b)

\(\frac{10}{33}\)

(c)

\(\frac{33}{10}\)

(d)

\(\frac{33}{20}\)

12.

The angle between the lines x²+ 4xy + y²= 0 is ______________

(a)

60⁰

(b)

15⁰

(c)

30⁰

(d)

45⁰

13.

Separate equation of lines for a pair of lines whose equation is x²+ xy -12y²= 0 are ______________

(a)

x + 4y = 0 and x + 3y = 0

(b)

2x - 3y = 0 and x - 4y = 0

(c)

x - 6y = 0 and x - 3y = 0

(d)

x + 4y = 0 and x - 3y = 0

14.

The equation of the straight line joining the origin to the point of intersection of y - x + 7 = 0 and y + 2x - 2 = 0 is ______________

(a)

3x + 4y = 0

(b)

3x - 4y = 0

(c)

4x - 3y = 0

(d)

4x + 3y = 0

15.

The equation x²+ kxy + y²- 5x - 7y + 6 = 0 represents a pair of straight lines then k = ______________

(a)

\(\frac{5}{3}\)

(b)

\(\frac{10}{3}\)

(c)

\(\frac{3}{2}\)

(d)

\(\frac{3}{10}\)

16.

The gradient of one of the lines of ax²+ 2hxy + by²= 0 is twice that of the other, then ______________

(a)

h²= ab

(b)

h = a + b

(c)

8h²= 9ab

(d)

9h²= 8ab

17.

The angle between the lines 2x - y + 5 = 0 and 3x + y + 4 = 0 is ______________

(a)

45⁰

(b)

30⁰

(c)

60⁰

(d)

90⁰

18.

The points (a, 0),(0, b) and (1, 1) will be collinear if ______________

(a)

a + b = 1

(b)

a + b = 2

(c)

\(\frac{1}{a}+\frac{1}{b}=1\)

(d)

a + b = 0

19.

If the points (2k, k) (k, 2k) and (k, k) enclose a triangle of area 18 sq units, then the centroid of the triangle is ______________

(a)

(8, 8)

(b)

(4, 4)

(c)

(3, 3)

(d)

(2, 2)

20.

The image of the point (3,8) in the line x+3y=7 is

(a)

(1,4)

(b)

(-1,-4)

(c)

(-4,-1)

(d)

(1,-4)

21.

The points (k + 1, 1), (2k + 1, 3) and (2k + 2, 2k) are collinear if ______________

(a)

k = -1

(b)

\(k=\frac{1}{2}\)

(c)

k = 3

(d)

k = 2

22.

The value \(\lambda\) for which the equation 12x²- 10xy + 2y²+11x-5y+\(\lambda\) = 0 represent a pair of straight lines is ______________

(a)

\(\lambda\) = 1

(b)

\(\lambda\) = 2

(c)

\(\lambda\) = 3

(d)

\(\lambda\) = 0

23.

If co-ordinate axes are the angle bisectors of the pair of lines ax²+ 2hxy + by²= 0 then ______________

(a)

a = b

(b)

h = 0

(c)

a + b = 0

(d)

a²+ b²= 0

24.

The equation of the bisectors of the angle between the lines represented by 3x²- 5xy + 4y²= 0 is ______________

(a)

3x²- 5xy - 3y²= 0

(b)

3x²+ 5xy + 4y²= 0

(c)

5x²- 2xy - 5y²= 0

(d)

5x²- 2xy + 5y²= 0

25.

If h²= ab, then the lines represented by ax²+ 2hx + by²= 0 are ______________

(a)

parallel

(b)

perpendicular

(c)

coincident

(d)

None

26.

The angle between the lines \((x^2+y^2)sin^2\alpha=(xcos\alpha-y\beta)^2\)

(a)

\(\alpha\)

(b)

\(2\alpha\)

(c)

\(\alpha+\beta\)

(d)

None

27.

Pair of lines perpendicular to the lines represented by ax²+ 2hxy + by²= 0 and through origin is ______________

(a)

ax²+ 2hxy + by²= 0

(b)

bx²+ 2hxy + ay²= 0

(c)

bx²- 2hxy + ay²= 0

(d)

bx²- 2hxy + ay²= 0

28.

The equation 3x²+ 2hxy + 3y²= 0 represents a pair of straight lines passing through the origin. The two lines are ______________

(a)

real and distinct if h²> 3

(b)

real and distinct if h²> 0

(c)

real and distinct h²> 6

(d)

real and distinct if h²- 9 = 0

29.

The condition that the slope of one of the lines represented by ax²+ 2hxy + by²= 0 is n times the slope of the other is ______________

(a)

4nh²= ab(1 + n)²

(b)

8h²= 9ab

(c)

4n = ab(1 + n)²

(d)

4nh²= ab

30.

The image of the point (1, 2) with respect to the line y = x is ______________

(a)

(-1, -2)

(b)

(2, 1)

(c)

(2, -1)

(d)

(2, 1)

31.

Which one of the following statements is false?

(a)

The image of a point \((\alpha\beta)\) about x-axis \((\alpha,-\beta)\)

(b)

The image of the line ax+by+c=0 about x-axis is ax-by+c=0

(c)

The image of a point \((\alpha,\beta)\) about y-axis \((-\alpha,\beta)\)

(d)

The image of the line ax+by+c=0 about y-axis is ax-by+c=0

32.

The co-ordinates of the foot of the perpendicular drawn from the point (2, 3) to the line 3x - y + 4 = 0 is ______________

(a)

\((\frac{1}{10},\frac{37}{10})\)

(b)

\((\frac{-1}{10},-\frac{37}{10})\)

(c)

\((\frac{-1}{10},\frac{37}{10})\)

(d)

\((\frac{37}{10},\frac{-1}{10})\)

33.

The lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0(a ≠ b ≠ c ≠ 1) are concurrent, then the value of \(\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=\) ______________

(a)

-1

(b)

(c)

(d)

abc

34.

Which one of the following statements in false?

(a)

A point \((\alpha,\beta)\) will lie on origin side of the line ax+by+c=0 if a\(\alpha\)+b\(\beta\)+c and c have the same sign

(b)

A point \((\alpha,\beta)\) will lie on non-origin side of the line ax+by+c=0 if a\(\alpha\)+b\(\beta\) +c and c have opposite sign

(c)

If \(\alpha=\frac{\pi}{2},p=0\) , then the equation xcos\(\alpha\)+ysin\(\alpha\)=p represents x-axis

(d)

If \(\alpha =0,p=0\), then the equation xcos\(\alpha\)+ysin\(\alpha\)=presents x-axis

35.

The distance of the point (2, 3) from the line 2x - 3y + 9 = 0 measured along the line 2x - 2y + 5 = 0 is ______________

(a)

\(\sqrt 2\)

(b)

\(2\sqrt 2\)

(c)

\(4\sqrt 2\)

(d)

36.

The lines x cos \(\alpha\) + y sin \(\alpha\) = p and xcos\(\beta\) + y sin\(\beta\) = q will be perpendicular if ______________

(a)

\(\alpha =\beta\)

(b)

\(\alpha-\beta=\frac{\pi}{2}\)

(c)

\(|\alpha-\beta|=\frac{\pi}{2}\)

(d)

\(\alpha-\beta=0\)

37.

The length of perpendicular from the origin to a line is 12 and the line makes an angle of 120° with the positive direction of y-axis. then the equation of line is ______________

(a)

\(x+y\sqrt 3=24\)

(b)

\(x+y=12\sqrt 2\)

(c)

x + y = 24

(d)

\(x+y=12\sqrt 3\)

38.

The equation of the straight line upon which the length of perpendicular from the origin is p and this normal makes an angle \(\theta\) with the positive direction of x-axis is ______________

(a)

x sin\(\theta\) + y cot\(\theta\) = p

(b)

x sin \(\theta\) + y cos \(\theta\) = p

(c)

x sin \(\theta\) + y tan \(\theta\) cos \(\theta\) = p

(d)

x cos \(\theta\) + y sin \(\theta\) = p

39.

The equation of the straight line which passes through the point (2, 4) and have intercept on the axes equal in magnitude but opposite in sign is ______________

(a)

x - y = 2

(b)

x - y + 2 = 0

(c)

x - y + 1 = 0

(d)

x - y - 1 = 0

40.

The equation of median from verten B of the triangle \(\triangle ABC\) the co-ordinates of whose vertices are A(-1, 6)B(-3, -9)C(5, -8) ___________

(a)

29x + 4y + 5 = 0

(b)

8x - 5y - 21 = 0

(c)

13x + 14y + 47 = 0

(d)

x + y -7 = 0

41.

The equation of the straight line bisecting the line segment joining the points (2, 4) and (4, 2) and making an angle of 45^o with positive direction of x-axis is ______________

(a)

x + y = 6

(b)

x - y = 0

(c)

x - y = 6

(d)

x + y = 0

42.

The equation of a line which makes an angle of 135° with positive direction of x-axis and passes through the point (1, 1) is ______________

(a)

x+y=2

(b)

x-y=0

(c)

\(2\sqrt {2x}-\sqrt {2y}=0\)

(d)

x-3y=0

43.

The equation of the bisectors of the angle between the co-ordinate axes are ______________

(a)

x+y=0

(b)

x-y=0

(c)

x\(\pm\)y=0

(d)

x=0

44.

The inclination to the x-axis and intercept on y-axis of the line \(\sqrt {2y}=x+2\sqrt 2\) ______________

(a)

\(30^0,\sqrt 2\)

(b)

30⁰,2

(c)

\(45^0,2\sqrt 2\)

(d)

45⁰,2

45.

If the straight line y = mx + c passes through the point (1, 2) and (-2, 4) then the value of m and c are ______________

(a)

\(\frac{8}{3},\frac{-2}{3}\)

(b)

\(\frac{-2}{3},\frac{8}{3}\)

(c)

\(\frac{2}{3},\frac{-8}{3}\)

(d)

\(\frac{-2}{3},\frac{-8}{3}\)

46.

Find the angle between the lines 3x²- 10xy - 3y² = 0 ______________

(a)

90°

(b)

45°

(c)

60°

(d)

30°

47.

The slope of the line joining A and B where A is (-1, 2) and B is the point of intersection of the lines 2x + 3y = 5 and 3x + 4y = 7 is ______________

(a)

-2

(b)

(c)

\(\frac{1}{2}\)

(d)

-\(\frac{1}{2}\)

48.

Find the nearest point on the line 3x + y = 10 from the origin is ______________

(a)

(2, 1)

(b)

(1, 2)

(c)

(3, 1)

(d)

(1,3)

49.

The lines x + 2y - 3 = 0 and 3x - y + 7 = 0 are ______________

(a)

parallel

(b)

neither parallel nor perpendicular

(c)

perpendicular

(d)

parallel as wellas perpendicular

50.

If(1, 3) (2,1) (9, 4) are collinear then a is ______________

(a)

\(\frac{1}{2}\)

(b)

(c)

(d)

-\(\frac{1}{2}\)

51.

The length of the perpendicular from origin to line is \(\sqrt{3}x-y+24=0\) is ______________

(a)

2\(\sqrt{3}\)

(b)

(c)

(d)

52.

The equating straight line with y-intercept -2 and inclination with x-axis is 135° is ______________

(a)

x + y - 2 = 0

(b)

y - x + 2 = 0

(c)

y + x + 2 = 0

(d)

none

53.

AB = 12 cm. AB slides with A on x-axis, B on y-axis respectively. Then the radius of the circle which is the locus of ΔAOB, where O is origin is ______________

(a)

(b)

(c)

(d)

54.

The locus of a moving point P(a cos³θ, a sin³θ) is ______________

(a)

\({ x }^{ \frac { 2 }{ 3 } }+{ y }^{ \frac { 2 }{ 3 } }={ a }^{ \frac { 2 }{ 3 } }\)

(b)

x²+ y²= a²

(c)

x + y = a

(d)

\({ x }^{ \frac { 3 }{ 2 } }+{ y }^{ \frac { 3 }{ 2 } }={ a }^{ \frac { 3 }{ 2 } }\)

55.

When h² = ab, the angle between the pair of straight lines ax² + 2hxy + by² = 0 is ______________

(a)

\(\frac\pi4\)

(b)

\(\frac\pi3\)

(c)

\(\frac\pi6\)

(d)

0^o

56.

If 7x² - 8xy +A = 0 represents a pair of perpendicular lines, the A is ______________

(a)

(b)

-7

(c)

-8

(d)

57.

The distance between the line 12x - 5y + 9 = 0 and the point (2, 1) is ______________

(a)

\(\pm\frac{28}{13}\)

(b)

\(\frac{28}{13}\)

(c)

\(-\frac{28}{13}\)

(d)

none of these

58.

A point equi-distant from the line 4x + 3y + 10 = 0, 5x -12y + 26 = 0 and 7x + 24y - 50 = 0 is ______________

(a)

(1, -1)

(b)

(1, 1)

(c)

(0, 0)

(d)

(0, 1)

59.

If the lines x + q = 0, y - 2 = 0 and 3x + 2y + 5 = 0 are concurrent, then the value of q will be ______________

(a)

(b)

(c)

(d)

60.

The value of \(\lambda\)for which the lines 3x + 4y = 5, 5x + 4y = 4 and \(\lambda\)x + 4y = 6 meet at a point is ______________

(a)

(b)

(c)

(d)

61.

The angle between the lines 2x - y + 3 = 0 and x + 2y + 3 = 0 is ______________

(a)

90°

(b)

60°

(c)

45°

(d)

30°

62.

Distance between the lines 5x + 3y - 7 = 0 and 15x + 9y + 14 = 0 is ______________

(a)

\(\frac{35}{\sqrt{34}}\)

(b)

\(\frac{1}{3\sqrt{34}}\)

(c)

\(\frac{35}{2\sqrt{34}}\)

(d)

\(\frac{35}{3\sqrt{34}}\)

63.

The figure formed by the lines ax ± by ± c = 0 is a ______________

(a)

rectangle

(b)

square

(c)

rhombus

(d)

none of these

64.

The equation of the line passing through (1, 5) and perpendicular to the line 3x -5y + 7 = 0 is ______________

(a)

5x + 3y - 20 = 0

(b)

3x - 5y + 7 = 0

(c)

3x - 5y + 6 = 0

(d)

5x + 3y + 7 = 0

65.

Slope of x-axis or a line parallel to x-axis is ______________

(a)

(b)

positive

(c)

negative

(d)

infinity

66.

If the points (a, 0) (0, b) and (x, y) are collinear, then ______________

(a)

\(\frac{x}{a}-\frac{y}{b}=1\)

(b)

\(\frac{x}{a}+\frac{y}{b}=1\)

(c)

\(\frac{x}{a}+\frac{y}{b}=-1\)

(d)

\(\frac{x}{a}+\frac{y}{b}=0\)

67.

The value of x so that 2 is the slope of the line through (2, 5) and (x, 3) is ______________

(a)

-1

(b)

(c)

(d)

68.

The locus of a point which moves such that it maintains equal distances from two fixed points is a ______________

(a)

straight line

(b)

line bisector

(c)

pair of straight lines

(d)

angle bisector

69.

The locus of a point which moves such that it maintains equal distance from the fixed point is a ______________

(a)

straight line

(b)

line bisector

(c)

circle

(d)

angle bisector

70.

One of the equation of the lines given by \(x^2+2xy \ cot \theta- y^2 = 0\) is

(a)

\(x-y\cot\theta =0\)

(b)

\(x+y\tan\theta =0\)

(c)

\(x\cos\theta+y(\sin\theta+1)=0\)

(d)

\(x\sin\theta+y(\cos\theta+1)=0\)

71.

\(\theta\) is acute angle between the lines x²- xy - 6y² = 0, then \(\frac{2\cos\theta+3\sin\theta}{4\sin\theta+5\cos\theta}\) is

(a)

(b)

\(-\frac{1}{9}\)

(c)

\(\frac{5}{9}\)

(d)

\(\frac{1}{9}\)

72.

If one of the lines given by 6x² - xy + 4cy² = 0 is 3x + 4y = 0, then c equals to

(a)

-3

(b)

-1

(c)

(d)

73.

The area of the triangle formed by the lines x² - 4y² = 0 and x = a is

(a)

2a²

(b)

\(\frac{\sqrt3}{2}a^2\)

(c)

\(\frac12a^2\)

(d)

\(\frac{2}{\sqrt3}a^2\)

74.

If the lines represented by the equations 6x²+ 41xy - 7y²= 0 make angles α and β with x-axis, then \(tan \ \alpha\tan \ \beta=\)

(a)

\(-\frac{6}{7}\)

(b)

\(\frac{6}{7}\)

(c)

\(-\frac{7}{6}\)

(d)

\(\frac{7}{6}\)

75.

If a vertex of a square is at the origin and its one side lies along the line 4x + 3y - 20 = 0, then the area of the square is

(a)

20 sq. units

(b)

16 sq. units

(c)

25 sq. units

(d)

4 sq.units

76.

If the two straight lines x + (2k -7)y + 3 = 0 and 3kx + 9y - 5 = 0 are perpendicular then the value of k is

(a)

k = 3

(b)

\(k=\frac13\)

(c)

\(k=\frac23\)

(d)

\(k=\frac32\)

77.

The y-intercept of the straight line passing through (1, 3) and perpendicular to 2x - 3y + 1 = 0 is

(a)

\(\frac{3}{2}\)

(b)

\(\frac{9}{2}\)

(c)

\(\frac{2}{3}\)

(d)

\(\frac{2}{9}\)

78.

The length of \(\bot\) from the origin to the line \(\frac{x}{3}-\frac{y}{4}=1,\) is

(a)

\(\frac{11}{5}\)

(b)

\(\frac{5}{12}\)

(c)

\(\frac{12}{5}\)

(d)

\(\frac{-5}{12}\)

79.

The image of the point (2, 3) in the line y = -x is

(a)

(-3, -2)

(b)

(-3, 2)

(c)

(-2, -3)

(d)

(3, 2)

80.

The point on the line 2x- 3y = 5 is equidistance from (1, 2) and (3, 4) is

(a)

(7, 3)

(b)

(4, 1)

(c)

(1, -1)

(d)

(-2, 3)

81.

The line (p + 2q)x + (p - 3q)y = p - q for different values of p and q passes through the point

(a)

\(\left(\frac{3}{5},\frac{5}{2}\right)\)

(b)

\(\left(\frac{2}{5},\frac{2}{5}\right)\)

(c)

\(\left(\frac{3}{5},\frac{3}{5}\right)\)

(d)

\(\left(\frac{2}{5},\frac{3}{5}\right)\)

82.

If the equation of the base opposite to the vertex (2, 3) of an equilateral triangle is x + y = 2, then the length of a side is

(a)

\(\sqrt{\frac{3}{2}}\)

(b)

(c)

\(\sqrt{6}\)

(d)

\(3\sqrt{2}\)

83.

Equation of the straight line perpendicular to the line x - y + 5 = 0, through the point of intersection the y-axis and the given line

(a)

x - y - 5 = 0

(b)

x + y - 5 = 0

(c)

x + y + 5 = 0

(d)

x + y + 10 = 0

84.

A line perpendicular to the line 5x - y = 0 forms a triangle with the coordinate axes. If the area of the triangle is 5 sq. units, then its equation is

(a)

\(x+5y\pm5\sqrt2=0\)

(b)

\(x-5y\pm5\sqrt2=0\)

(c)

\(5x+y\pm5\sqrt2=0\)

(d)

\(5x-y\pm5\sqrt2=0\)

85.

The equation of the line with slope 2 and the length of the perpendicular from the origin equal to \(\sqrt5\) is

(a)

x - 2y = \(\sqrt5\)

(b)

2x - y =\(\sqrt5\)

(c)

2x - y = 5

(d)

x - 2y - 5 = 0

86.

The intercepts of the perpendicular bisector of the line segment joining (1, 2) and (3, 4) with coordinate axes are

(a)

5, -5

(b)

5, 5

(c)

5, 3

(d)

5, -4

87.

The coordinates of the four vertices of a quadrilateral are (-2, 4), (-1, 2), (1, 2) and (2, 4) taken in order. The equation of the line passing through the vertex (-1, 2) and dividing the quadrilateral in the equal areas is

(a)

x + 1 = 0

(b)

x + y = 1

(c)

x + y + 3 = 0

(d)

x - y + 3 = 0

88.

Equation of the straight line that forms an isosceles triangle with coordinate axes in the I-quadrant with perimeter 4 + 2\(\sqrt{2}\) is

(a)

x + y + 2 = 0

(b)

x + y - 2 = 0

(c)

\(x+y-\sqrt{2}=0\)

(d)

\(x+y+\sqrt{2}=0\)

89.

The slope of the line which makes an angle 45^o with the line 3x- y = -5 are:

(a)

1, -1

(b)

\(\frac{1}{2},-2\)

(c)

\(1,\frac{1}{2}\)

(d)

\(2,-\frac{1}{2}\)

90.

Straight line joining the points (2, 3) and (-1, 4) passes through the point \((\alpha,\beta)\) if

(a)

\(\alpha+2\beta=7\)

(b)

\(3\alpha+\beta=9\)

(c)

\(\alpha+3\beta=11\)

(d)

\(3\alpha+\beta=11\)

91.

If the point (8, -5) lies on the locus \(\frac{x^2}{16}-\frac{y^2}{25}=k\), then the value of k is

(a)

(b)

(c)

(d)

92.

Which of the following point lie on the locus of 3x²+ 3y²- 8x - 12y + 17 = 0

(a)

(0, 0)

(b)

(-2, 3)

(c)

(1, 2)

(d)

(0, -1)

93.

Which of the following equation is the locus of (at², 2at)

(a)

\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)

(b)

\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)

(c)

x²+ y²= a²

(d)

y²= 4ax

94.

The equation of the locus of the point whose distance from y-axis is half the distance from origin is

(a)

x²+ 3y²= 0

(b)

x²- 3y²= 0

(c)

3x²+ y²= 0

(d)

3x²- y²= 0

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Answers Key:

Multiple Choice Question

94 x 1 = 94

(b)
\(\frac{x}{a}+\frac{y}{b}=1\)

(b)
y-axis

(b)
5x + 3y - 9 = 0

(b)
2y²= x

(a)
\(\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}\)

(b)
(-9, 6)

(b)
Opposite sides of the line 3x - 2y + 1 = 0

(d)
-3

10.

11.

12.

(a)
60⁰

13.

(d)
x + 4y = 0 and x - 3y = 0

14.

(d)
4x + 3y = 0

15.

(b)
\(\frac{10}{3}\)

16.

17.

(a)
45⁰

18.

19.

(a)
(8, 8)

20.

(b)
(-1,-4)

21.

(d)
k = 2

22.

(b)
\(\lambda\) = 2

23.

(b)
h = 0

24.

25.

26.

(b)
\(2\alpha\)

27.

28.

(b)
real and distinct if h²> 0

29.

(a)
4nh²= ab(1 + n)²

30.

(d)
(2, 1)

31.

(d)
The image of the line ax+by+c=0 about y-axis is ax-by+c=0

32.

33.

(a)
-1

34.

(d)
If \(\alpha =0,p=0\), then the equation xcos\(\alpha\)+ysin\(\alpha\)=presents x-axis

35.

36.

37.

(a)
\(x+y\sqrt 3=24\)

38.

(d)
x cos \(\theta\) + y sin \(\theta\) = p

39.

(b)
x - y + 2 = 0

40.

(b)
8x - 5y - 21 = 0

41.

(b)
x - y = 0

42.

(a)
x+y=2

43.

44.

(d)
45⁰,2

45.

(b)
\(\frac{-2}{3},\frac{8}{3}\)

46.

(a)
90°

47.

(d)
-\(\frac{1}{2}\)

48.

49.

(b)
neither parallel nor perpendicular

50.

(a)
\(\frac{1}{2}\)

51.

(d)
12

52.

53.

(a)
36

54.

(a)
\({ x }^{ \frac { 2 }{ 3 } }+{ y }^{ \frac { 2 }{ 3 } }={ a }^{ \frac { 2 }{ 3 } }\)

55.

(a)
\(\frac\pi4\)

56.

(b)
-7

57.

(b)
\(\frac{28}{13}\)

58.

59.

60.

(b)
1

61.

(a)
90°

62.

63.

64.

(a)
5x + 3y - 20 = 0

65.

(a)
0

66.

(b)
\(\frac{x}{a}+\frac{y}{b}=1\)

67.

(b)
1

68.

(b)
line bisector

69.

70.

\(x^{2}+2 x y \cot \theta-y^{2} =0 \)
\(x^{2}+x(2 y \cot \theta) +\left(-y^{2}\right)=0 \)
\(x =\frac{-2 y \cot \theta \pm \sqrt{4 y^{2} \cot ^{2} \theta+4 y^{2}}}{2} \)
\(x =-y \cot \theta \pm y \operatorname{cosec} \theta \)
\(x \sin \theta =-y \cos \theta-y \)
\(x \sin \theta+y(1+\cos \theta)=0 \)

71.

\(\tan \theta=\frac{2 \sqrt{b^{2}-a b}}{a+b}=\frac{2 \sqrt{\left(\frac{-1}{2}\right)^{2}+6}}{-5}\)
\(=\left|\frac{2 \sqrt{\frac{1}{4}+6}}{-5}\right|\)
\(=\frac{2\left(\frac{5}{2}\right)}{5}=1 \Rightarrow \theta=45^{\circ}\)
\(\frac{2 \cos \theta+3 \sin \theta}{4 \sin \theta+5 \cos \theta}=\frac{\frac{2}{\sqrt{2}}+\frac{3}{\sqrt{2}}}{\frac{4}{\sqrt{2}}+\frac{5}{\sqrt{2}}}=\frac{5}{9}\)

72.

Let the other line be ax + by = 0
(3x + 4y) (ax + by) = 6x² - xy + 4cy²
\(3 a =6 \)
\(a =2 \)
\(4 a+3 b =-1 \)
\(3 b =-9 \)
\(b =-3 \)
\(4 b =4 c \)
\(c =-3 \)

73.

\(x-2 y =0 \)
\(x+2 y =0 \)
\(x =a \)
\(\text {The points are }(0,0),\left(a, \frac{a}{2}\right),\left(a, \frac{-a}{2}\right)\)
\(\text {Area }=-\frac{1}{2} \text { (base } \times \text { height } \text { ) }\)
\(=\frac{1}{2}(a \cdot a)\)
\(\text {Area }=\frac{1}{2} a^{2}\)

74.

\(m_{1} m_{2}=\frac{a}{b}=\frac{6}{-7}\)

75.

Perpendicular distance from origin to the line is
4x + 3y - 20 = 0 is
\(\left(\frac{-20}{\sqrt{16+9}}\right)=\frac{20}{\sqrt{25}}=\frac{20}{5}=4 \text { units }\)
Area of the square = 4 \(\times\) 4 = 16 sq. units.

76.

\(\text { Slope of first line }=\frac{-1}{(2 k-7)}\)
\(\text { Slope of second line }=\frac{-3 k}{9}\)
\(\text { They are perpendicular } \mathrm{m}_{1} \mathrm{~m}_{2}=-1\)
\(\frac{3 k}{9(2 k-7)} =-1 \)
\(k =-3(2 k-7) \)
\(k =-6 k+21 \)
\(7 k =21 \)
\(k=3 \)

77.

2x- 3y +1 = 0
Perpendicular line is 3x + 2y + k = 0
This passes the through point (1, 3)
3 + 6 + k = 0
k = -9
Equation is 3x + 2y - 9 = 0
.'. y intercept is given by taking x = 0,
\(2 y=9, y=\frac{9}{2}\)

78.

Perpendicular distance from origin to the given line is
\(\frac{1}{\sqrt{\frac{1}{3^{2}}+\frac{1}{4^{2}}}}=\frac{1}{\sqrt{\frac{1}{9}+\frac{1}{16}}}=\frac{1}{\sqrt{\frac{16+9}{144}}}=\frac{12}{5}\)

79.

The required point is (-3, -2)

80.

Let (a, b) be on 2x - 3y= 5 = 2a - 3b = 5
It is equidistance from (1, 2) and (3, 4)
\(\sqrt{(a-1)^{2}+(6-2)^{2}}=\sqrt{(a-3)^{2}+(b-4)^{2}} \)
\((a-1)^{2}+(b-2)^{2}=(a-3)^{2}+(b-4)^{2} \)
\(a^{2}-2 a+1+b^{2}-4 b+4=a^{2}-6 a+9+b^{2} -8 b+16 \)
\(\begin{gathered} 4 a+4 b=20 \\ \Rightarrow {2 a+2 b=10}\\{2 a-3 b=5} \\ \hline 5 b=5 \end{gathered}\)
b = 1, a = 4
The point is (4, 1)

81.

\((p+2 q) x+(p-3 q) y-p+q=0 \)
\(p(x+y-1)+q(2 x-3 y+1)=0 \)
\(x+y=1 ...(1)\)
\(2 x-3 y=-1..(2)\)
\(\begin{array}{r} 3 \times(1) \Rightarrow 3 x+3 y=3 \\ (2) \Rightarrow 2 x-3 y=-1 \\ \hline 5 x=2 \end{array}\)
\(x=\frac{2}{5}, y=\frac{3}{5} \)
\(\therefore\left(\frac{2}{5}, \frac{3}{5}\right) \)

82.

Perpendicular distance from vertex to the opposite side is \(\frac{2+3-2}{\sqrt{1+1}}=\frac{3}{\sqrt{2}}\)
\(\frac{3}{\sqrt{2}}=\frac{\sqrt{3}}{2} a\)
\(\Rightarrow a =\frac{6}{\sqrt{6}} \)
\(\therefore a =\sqrt{6}\)

83.

x - y + 5 = 0;
Put x = 0, then y = 5
The point is (0, 5)
Any line perpendicular to x - y + 5 = 0 is
x + y + k = 0,
This passes through (0, 5)
k = -5
Required equation is x + y - 5 = 0

84.

\(5 x-y=0 \text { perpendicular line is } x+5 y+k=0\)
\(x \text { intercept is }-\mathrm{k} \text { and } \mathrm{y} \text { intercepts } \frac{-\mathrm{k}}{5}\)
\(\text { Area of the } \Delta=\frac{1}{2}(-\mathrm{k})\left(\frac{-\mathrm{k}}{5}\right)\)
\(5 =\frac{\mathrm{k}^{2}}{10} \text { (given) } \)
\(\mathrm{k}^{2} =50 \)
\(\mathrm{k} =\pm 5 \sqrt{2}\)
\(\text {Equation of the line is } x+5 y \pm 5 \sqrt{2}=0\)

85.

Let y = 2x + c be the required line.
Given perpendicular distance from origin'to this line is \(\sqrt5\)
\(\frac{c}{\sqrt{1+4}}=\sqrt{5}\)
c = 5
This required line is y - 2x + 5
2x - y + 5 = 0

86.

Equation of line joining (1, 2) and (3, 4) is
\(\frac{y-2}{4-2} =\frac{x-1}{3-1} \)
\(x-y+1 =0 \)
Any line perpendicular to this x + y + k = 0
This passes through midpoint of (1, 2) and (3, 4)
That is (2, 3)
k = -5, x+y-5 = 0
x intercept is 5, y intercept is 5.

87.

The point M is (0, 3)
The equation of line joining ('-1, 2) and (0, 3) is
\(\frac{y-2}{3-2} =\frac{x+1}{0+1} \)
\(x-y+3 =0\)

88.

\(\text {Perimeter }=4+2 \sqrt{2}\)
\(a+a+\sqrt{2} =4+2 \sqrt{2} \)
\(2 a+\sqrt{2} a =4+2 \sqrt{2} \)
\(\therefore a =2 \)
\(\text {Equation of line is } \frac{x}{2}+\frac{y}{2}=1\)
\(x+y-2=0\)

89.

\(\text { Slope of } 3 x-y+5=0 \text { is } \frac{-3}{-1}=3=m_{1}\)
Let m₂ be the slope of the second line
\(\text {Given } \tan \theta=\tan 45^{\circ}=1 \Rightarrow \frac{m_{1}-m_{2}}{1+m_{1} m_{2}}=\pm 1\)
\(\frac{3-m_{2}}{1+3 m_{2}}=1 \quad \frac{m_{2}-3}{1+3 m_{2}}=1\)
\(3-m_{2}=1+3 m_{2} \quad m_{2}-3=1+3 m_{2}\)
\(2=4 m_{2} \quad-2 m_{2}=4\)
\(\mathrm{m}_{2}=\frac{1}{2} \quad \mathrm{~m}_{2}=-2\)
\(\left(\frac{1}{2},-2\right)\)

90.

91.

\(\frac{\left(8^{2}\right)}{16}-\frac{(-5)^{2}}{25} =\mathrm{k} \)
\(\frac{64}{16}-\frac{25}{25} =\mathrm{k} \)
\(\mathrm{k} =4-1=3 \)

92.

\((1,2) \text { lies on } 3 x^{2}+3 y^{2}-8 x-12 y+17=0\)
\(\text { Because, } 3(1)^{2}+3(2)^{2}-8(1)-12(2)+17\)
\(=3+12-8-24+17=0\)

93.

\(\left(a t^{2}, 2 a t\right) \Rightarrow x=a t^{2}, \quad y=2 a t\)
\(y^{2} =4 a^{2} t^{2} \)
\(=4 a^{2}\left(\frac{x}{a}\right) \)
\(y^{2} =4 a x \)

94.

Let the point be (x, y)
Its distance from origin is \(\sqrt{x^{2}+y^{2}}\)
Given \(x =\frac{1}{2} \sqrt{x^{2}+y^{2}} \)
\(\Rightarrow 2 x =\sqrt{x^{2}+y^{2}} \)
\(4 x^{2} =x^{2}+y^{2} \)
\(3 x^{2}-y^{2}=0 \) is the required equation of the locus