Question

A point P moves such that the difference between its distance from the origin and from the axis of x is always a constant c. The locus of P is a

• A. Straight line having equal intercepts c on the axes.
• B. Circle having its centre at (0, -c/2) and passing through $(c\sqrt{2},-c/2)$
• C. Parabola with its vertex at (0, -c/2) and passing through $(c\sqrt{2},c/2)$
• D. none of these

Answer 1

Answer: (C)

Parabola with its vertex at (0, -c/2) and passing through $(c\sqrt{2},c/2)$

Let the coordinates of $P$ be $(h,k)$

Then $\sqrt{h^2+k^2}-k=c$

$\Rightarrow$ $h^2+k^2={(k+c)}^2$

$\Rightarrow$ $h^2=2c(k+c/2)$

Locus of (h, k) is $x^2=2c(y+c/2)$

which represents a parabola with vertex (0, -c/2) and passing through $(c\sqrt{2},c/2)$.