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Mathematics

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

Coordinate Geometry

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Answer

By distance formula,

d = (x2−x1)2+(y2−y1)2\sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Given,

AB = AC.

⇒(3−0)2+(p−2)2=(p−0)2+(5−2)2⇒32+p2+4−4p=p2+32⇒9+p2+4−4p=p2+32\Rightarrow \sqrt{(3 - 0)^2 + (p - 2)^2} = \sqrt{(p - 0)^2 + (5 - 2)^2} \\[1em] \Rightarrow \sqrt{3^2 + p^2 + 4 - 4p} = \sqrt{p^2 + 3^2} \\[1em] \Rightarrow \sqrt{9 + p^2 + 4 - 4p} = \sqrt{p^2 + 3^2}

On squaring both sides,

⇒p2+13−4p=p2+9⇒p2−p2+4p=13−9⇒4p=4⇒p=1.\Rightarrow p^2 + 13 - 4p = p^2 + 9 \\[1em] \Rightarrow p^2 - p^2 + 4p = 13 - 9 \\[1em] \Rightarrow 4p = 4 \\[1em] \Rightarrow p = 1.

Hence, p = 1.

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