Question

The equation of the circle described on the chord $3x+y+5=0$ of the circle $x^2+y^2=16$ as diameter is :

• A. $x^2+y^2+3x+y-11=0$
• B. $x^2+y^2+3x+y+1=0$
• C. $x^2+y^2+3x+y-2=0$
• D. $x^2+y^2+3x+y-22=0$

Answer 1

Answer: (A)

Given circle is $x^2+y^2-16=0$
Eqn. of chord say AB of given circle is
$3x+y+5=0$.
Equation of required circle is
$x^2+y^2-16+\lambda \left(3x+y+5\right)=0$
$\Rightarrow x^2+y^2+\left(3\lambda \right)x+\left(\lambda \right)y+5\lambda -16=0...\left(1\right)$
Centre $C=\left(\frac{-3\lambda }{2},\frac{-\lambda }{2}\right)$
If line AB is the diameter of circle $\left(1\right)$, then
$C\left(\frac{-3\lambda }{2},\frac{-\lambda }{2}\right)$ will lie on line AB.
$3\left(\frac{-3\lambda }{2}\right)+\left(\frac{-\lambda }{2}\right)+5=0$
$\Rightarrow -\frac{9\lambda -\lambda }{2}+5=0\Rightarrow \lambda =1$
Hence, required eqn of circle is
$x^2+y^2+3x+y+5-16=0$
$\Rightarrow x^2+y^2+3x+y-11=0$