Question

The equation of a circle which passes through the three points $(3,0),(1,-6),(4,-1)$ is-

• A. $2x^2+2y^2+5x-11y+3=0$
• B. $x^2+y^2-5x+11y-3=0$
• C. $x^2+y^2+5x-11y+3=0$
• D. $2x^2+2y^2-5x+11y-3=0$

Answer 1

The correct option is D $2x^2+2y^2-5x+11y-3=0$

Given three points are $A(3,0),B(1,-6),C(4,-1)$

Equation of the circle with $A,B$ as end point of diameter is $(x-3)(x-1)+y(y+6)=0$

$S:x^2+y^2-4x+6y+3=0$

Equation of line $AB$ is $3x-y-9=0$

Any circle passing through $A,B$ is of the form $S+\lambda P=0$ where $P$ is equation of line $AB$.

$\Rightarrow x^2+y^2-4x+6y+3+\lambda (3x-y-9)=0$

But, $C(4,-1)$ lies on the above circle.

$\Rightarrow -2+4\lambda =0$

$\Rightarrow \lambda =\frac{1}{2}$

$\therefore$ Equation of the circle passing through the given points is $2x^2+2y^2-5x+11y-3=0$
Hence, option D.