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Mathematics

If A = (x, -7), B = (2, 5) and AB = 13 units, find x.

Coordinate Geometry

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Answer

Distance between 2 points (x1, y1) and (x2, y2) = (x2−x1)2+(y2−y1)2\sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Distance between A = (x, -7) and B = (2, 5) =

⇒13=(x−2)2+(5−(−7))2⇒132=(x−2)2+(5+7)2⇒169=x2+4−4x+122⇒169=x2+4−4x+144⇒x2+4−4x+144−169=0⇒x2−4x−21=0⇒x2−7x+3x−21=0⇒x(x−7)+3(x−7)=0⇒(x−7)(x+3)=0⇒(x−7)=0 or (x+3)=0⇒x=7 or x=−3\Rightarrow 13 = \sqrt{(x - 2)^2 + (5 - (-7))^2}\\[1em] \Rightarrow 13^2 = (x - 2)^2 + (5 + 7)^2\\[1em] \Rightarrow 169 = x^2 + 4 - 4x + 12^2\\[1em] \Rightarrow 169 = x^2 + 4 - 4x + 144\\[1em] \Rightarrow x^2 + 4 - 4x + 144 - 169 = 0\\[1em] \Rightarrow x^2 - 4x - 21 = 0\\[1em] \Rightarrow x^2 - 7x + 3x - 21 = 0\\[1em] \Rightarrow x(x - 7) + 3(x - 7) = 0\\[1em] \Rightarrow (x - 7)(x + 3) = 0\\[1em] \Rightarrow (x - 7) = 0 \text{ or } (x + 3) = 0\\[1em] \Rightarrow x = 7 \text{ or } x = -3\\[1em]

Hence, the value of x = 7 or -3.

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