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 `=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

\[\Rightarrow \sqrt{\left( 3 - 0 \right)^2 + \left( p - 2 \right)^2} = \sqrt{\left( p - 0 \right)^2 + \left( 5 - 2 \right)^2}\]                           (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.