# Show that all the chords of the curve 3x2-y2-2x+4y=0 which subtend a right angle at the origin pass through a fixed point. Find that point.

Harsh Patodia IIT Roorkee
9 years ago
Let y=mx + c be the equation of chord
Homogenize the equation of curve 3x2-y2-2x+4y=0 by putting 1= (y-mx)/c
Equation becomes 3x2-y2-2x(y-mx)/c+4y(y-mx)/c=0
Homogenization gives the equation of pair lines from origin on the given curve
Since the angle subtended at origin by the chord is 900 . Angle between pair of line is 900.
Condition for that is sum of coefficient of x2 and y2 in the homogenized equation is equal to 0.
Hence it will give after simpllication c + m +2=0
On rearranging -2= 1.m + C ….….….…....(1)
Compariing with y = mx + c
(1) passes through a fixed point (1,-2)
RAHUL KUMAR
13 Points
4 years ago
as we can observe that curve is passing through origin.
so find out the slope of tangent at origin.
Differentiate curve eqn w.r.t  “x”
6*x – 2*y*dy/dx – 2 + 4*dy/dx = 0
put (x,y) = (0,0)
slope of tangent = dy/dx = ½
:: since chord is subtending 900 on curve at origin, hence it will be perpendicular to the tangent also.
slope of chord = -2
now we have point (0,0) and slope = -2
eqn of chord :: $y=-2*x$
$y +2*x =0$
to find correct answer, satisfy given points in options