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Questions:

Multiple Choice Question

50 x 1 = 50

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.

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)

47.

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)

48.

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)

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

49.

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

50.

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

50 x 1 = 50

(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.

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

47.

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)

48.

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

49.

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

50.

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