# 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. - Mathematics

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.

#### Solution 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

#### Solution 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.

Concept: Distance Formula
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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.2 | Q 37 | Page 17
RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.2 | Q 44 | Page 17