Question

Consider a triangle A whose two sides lie on the x - axis and the line x + y + 1 = 0. If the orthocenter of A is (1, 1), then the equation of the circle passing through the vertices of the triangle ▲ is x ^ 2 + y ^ 2 - 3x + y = 0 x ^ 2 + y ^ 2 + x + 3y = 0 x ^ 2 + y ^ 2 + 2y - 1 = 0 x ^ 2 + y ^ 2 + x + y = 0 C D A B

Answer 1

Correct option is (b) x² + y² + x + 3y = 0

As we know mirror image of orthocenter lie on circumcircle.

∴ Image of (1,1) in x−axis is (1,−1)

Image of (1,1) in x + y + 1 = 0 is given by

\(\frac{x-1}{1}=\frac{x-1}{1}=\frac{2(1+1+1)}{1^2+1^2}\)

⇒ x = −2 and y = −2

∴ Image of (1,1) in x + y + 1 = 0 is (−2,−2).

and intersection of x− axis and line x + y + 1 = 0 is (−1,0).

∴ The required circle will be passing through the points (1,–1),(−2,−2) and (−1,0),

Hence, the equation of circle is x² + y² + x + 3y = 0