Question

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.

If a point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), then find the value of p.

Answer 1

The given points are A(0, 2), B(3, p) and C(p, 5).

It is given that A is equidistant from B and C.

∴ AB = AC

⇒ AB² = AC²

⇒ (3 − 0)² + (p − 2)² = (p − 0)² + (5 − 2)²

⇒ 9 + p² + 4 − 4p = p² + 9

⇒ 4 − 4p = 0

⇒ 4p = 4

⇒ p = 1

Thus, the value of p is 1.

Length of AB ${\displaystyle =\sqrt{{(3-0)}^2+{(1-2)}^2}=\sqrt{3^2+{(-1)}^2}=\sqrt{9+1}=\sqrt{10}units}$

Answer 2

It is given that A(0, 2) is equidistant from the points B(3, p) and C(p, 5).
∴ AB = AC

$$\Rightarrow \sqrt{{(3-0)}^2+{(p-2)}^2}=\sqrt{{(p-0)}^2+{(5-2)}^2}$$                           (Distance formula)

Squaring on both sides, we get

$$9+p^2-4p+4=p^2+9$$
$$\Rightarrow -4p+4=0$$
$$\Rightarrow p=1$$

Thus, the value of p is 1.