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……………………?

Answer 1

The distance of any point from x-axis is the value of ordinate.
So now if P(x, y) is a point, then it's distance from x-axis is d(say).

$d = \left \{ {{y : y \geq 0} \atop {-y:y < 0}} \right.$

Also the distance of any point P(x,y) is given by square-root of sum of squares of abscissa and ordinate, So

$d_o_r_i_g_i_n = \sqrt{ x^{2} + y^{2} }$

Assuming $y \geq 0$

$\sqrt{ x^{2} + y^{2} } - y = c \\ \\ \sqrt{ x^{2} + y^{2} } = y + c \\ \\ x^{2} + y^{2} = (y + c) ^2 \\ \\ x^{2} + y^{2} = y^{2} + c^{2} + 2(c) (y) \\ \\ x^{2} - 2(c)(y) - c^{2} = 0$

Thus,
$Locus = \left \{ {{ x^{2} - 2(y)(c) - c^{2} : y \geq 0} \atop { x^{2} + 2(c)(y) - c^{2} : y < 0}} \right.$