Question
A point moves such that the square of its distance from a straight line is equal to the difference between the square of its distance from the centre of a circle and the square of the radius of the circle. The locus of the point is :
A.
a straight line at right angle to the given line
B.
a circle concentric with the given circle
C.
a parabola with its axis parallel to the given line
D.
a parabola with its axis perpendicular to the given line
Answer :
a parabola with its axis perpendicular to the given line
Solution :
Let the given line be the $$y$$-axis and the circle to have the equation $${x^2} + {y^2} + 2gx + 2fy + c = 0$$
then according to given condition
$$\eqalign{ & {x^2} = {\left( {x + g} \right)^2} + {\left( {y + f} \right)^2} - \left( {{g^2} + {f^2} - c} \right) \cr & \Rightarrow {\left( {y + f} \right)^2} = - 2g\left( {x - \frac{{{f^2} - c}}{{2g}}} \right), \cr} $$
which represents a parabola with its axis $$ \bot $$ to $$y$$-axis.
Let the given line be the $$y$$-axis and the circle to have the equation $${x^2} + {y^2} + 2gx + 2fy + c = 0$$
then according to given condition
$$\eqalign{ & {x^2} = {\left( {x + g} \right)^2} + {\left( {y + f} \right)^2} - \left( {{g^2} + {f^2} - c} \right) \cr & \Rightarrow {\left( {y + f} \right)^2} = - 2g\left( {x - \frac{{{f^2} - c}}{{2g}}} \right), \cr} $$
which represents a parabola with its axis $$ \bot $$ to $$y$$-axis.