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Geometry Level 3

The co-ordinates of the circumcenter of the triangle formed by the lines 2 x 2 + 3 x y − 2 y 2 − 6 x + 8 y − 8 = 0 , x = y 2x^{2}+3xy-2y^{2}-6x+8y-8=0 , x=y is ( a , b ) (a,b) . Find a − b a-b .

Source: K V P Y \color{#EC7300}{KVPY} .


The answer is 0.

Shivam Jadhav by Shivam Jadhav , 22, India

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2 solutions

Sid Rana
Apr 18, 2017

Circumcentre lies on x=y a-b=0

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Rajen Kapur
Oct 15, 2015

As ( 2 x − y + 2 ) ( x + 2 y − 4 ) = 0 (2x - y +2)(x + 2y -4)=0 are two perpendicular lines making a right-angled triangle with the line x = y x = y , the hypotenuse. In a right-angled triangle the circumcenter lies on the hypotenuse, hence a = b, or (a - b)= 0.

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