Straight Line and Pair of Straight Lines 1 Question 64
64. Two equal sides of an isosceles triangle are given by the equations $7 x-y+3=0$ and $x+y-3=0$ and its third side passes through the point $(1,-10)$. Determine the equation of the third side.
$(1984,4 M)$
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Solution:
- The equation of any line passing through $(1,-10)$ is $y+10=m(x-1)$.
Since, it makes equal angles, say $\theta$, with the given lines, therefore
$$ \tan \theta=\frac{m-7}{1+7 m}=\frac{m-(-1)}{1+m(-1)} \quad \Rightarrow \quad m=\frac{1}{3} \text { or }-3 $$
Hence, the equations of third side are
$$ \begin{aligned} & \quad y+10=\frac{1}{3}(x-1) \text { or } \quad y+10=-3(x-1) \\ & \text { i.e. } \quad x-3 y-31=0 \text { or } 3 x+y+7=0 \end{aligned} $$
