Question

The locus of the centre of a circle passing through a point and touching a line is

• A. a straight line
• B. an ellipse
• C. a parabola
• D. a hyperbola

Answer 1

The correct option is C a parabola


Let the point be $(\alpha ,\beta )$ and the line be $Ax+By+C=0$
Assume centre of circle be $(x_1,y_1)$, then
$\sqrt{(x_1-\alpha {)}^2+(y_1-\beta {)}^2}=\left|\frac{A{x}_1+B{y}_1+C}{\sqrt{A^2+B^2}}\right|$
$\Rightarrow (x_1-\alpha {)}^2+(y_1-\beta {)}^2={\left(\frac{A{x}_1+B{y}_1+C}{\sqrt{A^2+B^2}}\right)}^2$
Clearly it is an equation of parabola.