Question

Two equal sides of an isosceles triangle are given by $7x-y+3=0$ and $x+y-3=0$ and the third side passes through the point $(1,10),$ the slope $m$ of the third side is given by

• A. $3m^2-1=0$
• B. $m^2+1=0$
• C. $3m^2+8m-3=0$
• D. $m^2-3=0$

Answer 1

The correct option is C $3m^2+8m-3=0$

The given equations are $7x-y+3=0\to \left(1\right)$ and $x+y-3=0\to \left(2\right)$

Let the slope of the equation $\left(1\right)$ be $m_1$ and equation$\left(2\right)$ be $m_2$

$\therefore m_1=7$ and $m_2=-1$

Let the slope of the third side be $m_3=m$

Angle between the two sides of a triangle having slopes $m_1$ and $m_2$ is given by $\frac{m_1-m_2}{1+m_1{m}_2}$

Angle between line $\left(1\right)$ and line $\left(3\right)$ is equal to the angle between line $\left(2\right)$ and line $\left(3\right)$ because the given triangle is an isoceles triangle.

So, $\frac{7-m}{1+7m}=\frac{m+1}{1-m}$

Solving the above equation gives $3m^2+8m-3=0$