question_answer

A point moves such that the square of its distance from a straight line is equal to the difference between the square of it 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

Correct Answer: D

Solution :

[d] Let the given line be the y-axis and the circle to Have the eq. \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] then according to given condition \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] then according to given condition \[{{x}^{2}}={{(x+g)}^{2}}+{{(y+f)}^{2}}-({{g}^{2}}+{{f}^{2}}-c)\] \[\Rightarrow {{(y+f)}^{2}}=-2g\left( x-\frac{{{f}^{2}}-c}{2g} \right),\] Which represents a parabola with its axis \[\bot \] to y-axis.