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

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

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)

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

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)

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

(a)

-3

(b)

-1

(c)

3

(d)

1

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)

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

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

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

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

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

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

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

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

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)

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

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)

\(\alpha-\beta=0\)

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)

4

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

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)

1

(c)

0

(d)

abc

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)

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

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

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)

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

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

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

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

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

(a)

parallel

(b)

perpendicular

(c)

coincident

(d)

None

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

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

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

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

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)

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)

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

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

(a)

45⁰

(b)

30⁰

(c)

60⁰

(d)

90⁰

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

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)

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

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

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

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

(a)

60⁰

(b)

15⁰

(c)

30⁰

(d)

45⁰

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)

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

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)

1

(d)

2

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

(a)

1

(b)

-1

(c)

3

(d)

-3

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

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)

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)

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

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

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

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

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 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)

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