Two sides of an isosceles triangle are given by the equations $7 x - y + 3 = 0 a n d x + y - 3 = 0 .$ If its third side passes through the point $(1, - 10),$ then its equations are

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$\begin{aligned} x - 3 y - 7 = 0 or 3 x + y - 31 = 0 \end{aligned}$

$\begin{array}{r} x - 3 y - 31 = 0 or 3 x + y - 7 = 0 \end{array}$

$x - 3 y - 31 = 0 or 3 x + y + 7 = 0$

none of these

answer is C.

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Detailed Solution

Any line through $(1, - 10) i s (y + 10) = m (x - 1)$

Since it makes equal angles, say $θ,$ with the given lines, we have

$\tan θ = \frac{m - 7}{1 + 7 m} = - \frac{m - (- 1)}{1 + m (- 1)}$

This gives $m = 1 / 3 or - 3 .$

Hence the equation of third side is

$y + 10 = \frac{1}{3} (x - 1) y + 10 = - 3 (x - 1) i . e, x - 3 y - 31 = 0 or 3 x + y + 7 = 0$

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