Question

The equation of two equal sides of an isosceles triangle are $7x-y+3=0$ and $x+y-3=0.$ If the third side passes through $(1,-10)$ and its slope is negative, then the equation of the third side is

• A. $3x+y+7=0$
• B. $2x+y+8=0$
• C. $3x+y-7=0$
• D. $x-3y-31=0$

Answer 1

The correct option is A $3x+y+7=0$

Let $m$ be the slope of the third line.
Equation of the line with slope $m$ and passing through $(1,-10)$ is
$y+10=m(x-1)$
Since the third line makes equal angle with the given two lines,
$\left|\frac{m-7}{1+7m}\right|=\left|\frac{m-(-1)}{1-m}\right|$
Taking positive signs, we get
$m^2+1=0$ (Not possible)

Hence, $\frac{m-7}{1+7m}=-\left(\frac{m+1}{1-m}\right)$
$\Rightarrow 3m^2+8m-3=0$
$\Rightarrow (3m-1)(m+3)=0$
$\Rightarrow m=\frac{1}{3}$ or $m=-3$
But $m<0$
$\therefore$ Equation of the third line is $3x+y+7=0$