Question

If the line $y=mx+a$ meets the parabola $y^2=4ax$ at two points whose abscissas are $x_1$ and $x_2,$ then $x_1+x_2=0$ if

• A. $m=-1$
• B. $m=1$
• C. $m=2$
• D. $m=-\frac{1}{2}$

Answer 1

The correct option is C $m=2$

Given line is $y=mx+a\cdots \left(1\right)$

$y^2=4ax\cdots \left(2\right)$

Substitute $y$ from equation $\left(1\right)$ into equation $\left(2\right)$

$(mx+a{)}^2=4ax$

$\Rightarrow m^2{x}^2+2amx+a^2=4ax$

$\Rightarrow m^2{x}^2+(2am-4a)x+a^2=0$

The two roots of the above quadratic equation are $x_1$ and $x_2$.

For $x_1+x_2=0,$ we have

Sum of roots $=\frac{-\text{Coefficient of}x}{\text{Coefficient of}{x}^2}=-\frac{2am-4a}{m^2}$

$x_1+x_2=-\frac{2am-4a}{m^2}$

$\therefore -\frac{2am-4a}{m^2}=0$

$\Rightarrow 4a-2am=0$

$\Rightarrow 4a=2am$

$\Rightarrow m=\frac{4a}{2a}$

$\Rightarrow m=2$