If $B (1, 3)$ is equidistant from $A (6, - 1)$ and $C (x, 8)$ , then $x =$

3 or - 5

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- 3 or 5

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- 3 or - 5

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3 or 5

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Solution

The correct option is B $- 3 or 5$
Given Points

$A (6, - 1) B (1, 3) C (x, 8)$

Distance from $A$ to $B$

$A B = \sqrt{(6 - 1)^{2} + (3 + 1)^{2}}$

$A B = \sqrt{5^{2} + 4^{2}}$

$A B = \sqrt{25 + 16}$

$A B = \sqrt{41}$

So the Distance from $B$ to $C$ is $\sqrt{41}$

$⟹ B C = \sqrt{(x - 1)^{2} + (8 - 3)^{2}}$

$⟹ \sqrt{41} = \sqrt{x^{2} - 2 x + 1 + 25}$

$⟹ 41 = x^{2} - 2 x + 26$

$⟹ x^{2} - 2 x - 15 = 0$

$⟹ (x + 3) (x - 5) = 0$

$⟹ x = - 3, 5$

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