Straight Line and Pair of Straight Lines 4 Question 3

3. Show that all chords of curve 3x2y22x+4y=0, which subtend a right angle at the origin pass through a fixed point. Find the coordinates of the point.

(1991,4 M)

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Answer:

Correct Answer: 3. (1,2)

Solution:

  1. The given curve is

3x2y22x+4y=0

Let y=mx+c be the chord of curve (i) which subtend right angle at origin. Then, the combined equation of lines joining points of intersection of curve (i) and chord y=mx+c to the origin, can be obtained by the equation of the curve homogeneous, i.e.

3x2y22xymxc+4yymxc=03cx2cy22xy+2mx2+4y24mxy=0(3c+2m)x22(1+2m)y+(4c)y2=0

Since, the lines represented are perpendicular to each other.

Coefficient of x2+ Coefficient of y2=0

3c+2m+4c=0

c+m+2=0

On comparing with y=mx+c

y=mx+c passes through (1,2).



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