Its all fixed

Geometry Level 5

All the chords of the curve 3 x 2 − y 2 − 2 x + 4 y = 0 3x^{2}-y^{2}-2x+4y=0 , which subtend a right angle at the origin pass through a fixed point. If the coordinate of the point is ( a , b ) (a,b) , find 16 a + b 16a+b .

This problem is a part of the set JEE Fest .


The answer is 14.

Dheeman Kuaner by Dheeman Kuaner , 21, India

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2 solutions

James Wilson
Nov 2, 2017

I solved it by considering the two points as ( x 1 , y 1 ) (x_1,y_1) and ( − k y 1 , k x 1 ) (-ky_1,kx_1) . Then I substituted each into the equation of the hyperbola, and took k = 6 , − 1 / 6 k=6, -1/6 . This gave me two pairs of points for which I formed the equations of their lines. The intersection point had to be none other than the fixed point.

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Dheeman Kuaner
Jun 30, 2017

Many people may not be familiar with homogenization.Refer here for introduction to homogenisation .

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Thanks for sharing this solution. I had never seen this technique before.

James Wilson - 3 years, 7 months ago

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Your welcome

Dheeman Kuaner - 3 years, 7 months ago

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