Question

All chords of the curve 3x2 – y2 –2x + 4y = 0 that subtends a right angle at the origin, pass through a fixed point whose co-ordinate is –

Solution

Correct option is

(1, – 2)

Let the equation of chord be y = mx + c. Combined equation of lines joining the point of intersection with origin is

                  

i.e., x2. (3c + 2m) – y2. (c  – 4) – 2xy. (1 + 2m) = 0

These lines will be mutually perpendicular if

                 3c + 2m – c + 4 = 0.

 that means the chord

 y =  mx + c is always pass through the point (1, –2).

SIMILAR QUESTIONS

Q1

The line x + y = 1 meets the line represented by the equation y3 –xy2 – 14x2y + 24x3 = 0 at the points ABC. If O is the origin, then

OA2 + OB2 + OC2 is equal to    

Q2

If the area of the triangle formed by the pair of lines 8x2 – 6xy + y2 = 0 and the line 2x + 3y = a is 7, then a is equal to

 

Q3

If p is length of the perpendicular from the origin on the line  are in A.P. then ab is equal to  

Q4

If two of the lines given by the equation ax3 – 9yx2 – y2x + 4y3 = 0 are perpendicular then a is equal to

Q5

The number of straight lines equidistant from three non-collinear points in the plane of the points is equal to

Q6

A line bisecting the ordinate PN of a point P(at2, 2at), t > 0 on the parabola y2 = 4ax, a > 0 is drawn parallel to the axis to meet the curve at Q. If NQ and the tangent at the point P meet at, T, then the coordinates of point T is (where point N lie on the axis)

Q7

If the pair of line  intersect on x-axis, then α is equal to –

Q8

If the area of the rhombus enclosed by the lines  be 2 square units, then

Q9

If a2 + b2 – c2 – 2ab = 0 then the point of concurrency of family of straight lines ax + by + c = 0lies on the line –

Q10

In a triangle ABC,  and point A lies on line y = 2x + 3 where  Area of  is such that [∆] = 5. Possible co-ordinates of ∆ is/are –