Question

The radical centre of three circles taken in pairs described on the sides of a triangle ABC as diameters is the

• A. centroid of the $\Delta ABC$
• B. oncentre of the $\Delta ABC$
• C. circumcentre of the $\Delta ABC$
• D. orthocentre of the $\Delta ABC$

Answer 1

The correct option is D orthocentre of the $\Delta ABC$

Let $A(a,0),B(-a,0),C(\alpha ,\beta )$

The equation of circles with $AB,BCandCA$ as diameters are

$(x–a)(x+a)+(y–0)(y+0)=0⟹x^2+y^2=a^2$

$x^2+y^2+(a-\alpha )x-\beta y-a\alpha =0$

$x^2+y^2-(a+\alpha )x-\beta y+a\alpha =0$

Radical axis of $S_2$ and $S_3$ is given by $S_2–S_3=0$

$⟹2ax–2a\alpha =0$

$⟹x=\alpha asa\ne 0$

But, $x=\alpha$ is the altitude through C. Similarly, other R.A will be altitudes.

Radical Center will be orthocenter.