Question

A circle touches a given straight line and cuts off a constant length $2d$ from another straight line perpendicular to the first straight line. The locus of the centre of the circle is?

• A. $y^2-4x^2=4d^2$
• B. $x^2+y^2=d^2$
• C. $xy=d^3$
• D. none of these

Answer 1

The correct option is A $y^2-4x^2=4d^2$

Let the center of circle be $(h,k)$, it touches the x-axis and cut a constant length $2d$ on y-axis.

We have the equation of the circle:

$(x-h{)}^2+(y-k{)}^2=k^2$

Circle intersection with y-axis $x=0$;

$(-h{)}^2+(y-k{)}^2=k^2$

$y^2-2hk+h^2=0$

$y=\frac{k\pm \sqrt{k^2-4h^2}}{2}$

Now equate given length with difference of intersecting coordinates

$\sqrt{k^2-4h^2}=2d$

$k^2-4h^2=(2d{)}^2=4d^2$

$h\to x,k\to y$

$\therefore y^2-4x^2=4d^2$.

It's a hyperbola.