Question

A Circle touches x – axis and cuts off a chord of length 2/from y–axis. The locus of the centre of the circle is

• A. A straight line
• B. A circle
• C. An ellipse
• D. A hyperbola

Answer 1

The correct option is D A hyperbola

If the circle $x^2+y^2+2gx+2fy+c=0$ touches the x-axis,
Then $-f=\sqrt{g^2+f^2-c}\Rightarrow g^2=c\dots \dots \dots \left(i\right)$
And cuts a chord of length 2l from y - axis
$\Rightarrow 2\sqrt{f^2-c}=2l\Rightarrow f^2-c=l^2\dots \dots \dots \dots ..\left(ii\right)$
Subtracting (i) from (ii), we get $f^2-g^2=l^2$.
Hence the locus is $y^2-x^2=l^2$, which is a rectangular hyperbola.