Question

If the intercepts of the variable circle on the $x$ and $y$-axis are $2$ units and $4$ units respectively, then the locus of the centre of the variable circle is

• A. $x^2-y^2+3=0$
• B. $2y^2-x^2+4=0$
• C. $x^2-2y^2+4=0$
• D. $y^2-x^2+3=0$

Answer 1

The correct option is A $x^2-y^2+3=0$

Let equation of variable circle is $x^2+y^2+2gx+2fy+c=0$, which is centred at $(-g,-f)$

Given that $2\sqrt{g^2-c}=2$ and $2\sqrt{f^2-c}=4$
$\Rightarrow g^2-c=1$ and $f^2-c=4$
$\Rightarrow f^2-g^2=3$
$\Rightarrow (-f{)}^2-(-g{)}^2=3$
Hence, locus of centre is $y^2-x^2=3$