If the distance of the point P (x, y) from A (0, a) is a + y, then x2 equals to:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4ay
Write the distance PA with the distance formula, set it equal to \(a+y\), and square both sides to remove the square root.
Distance formula: \(PA=\sqrt{(x-0)^{2}+(y-a)^{2}}\), and this equals \(a+y\)
Squaring both sides: \(x^{2}+y^{2}-2ay+a^{2}=a^{2}+2ay+y^{2}\)
The \(y^{2}\) and \(a^{2}\) terms appear on both sides and cancel: \(x^{2}-2ay=2ay\)
So \(x^{2}=4ay\): the points P form a parabola with its vertex at the origin