Question

If the line $y=mx+c$ touches the parabola $y^{2}=12\left(x + 3\right)$ exactly for one value of $m$ $\left(m > 0\right)$ , then the value of $\frac{c + m}{c - m}$ is equal to

Answer 1

Equation of the tangent to $y^{2}=12\left(x + 3\right)$ with slope $m$ is
$y=m\left(x + 3\right)+\frac{3}{m},$ which is same as $y=mx+c$
$\Rightarrow c=3m+\frac{3}{m}$
$\Rightarrow 3m^{2}-mc+3=0$ which is quadratic in $‘m’$
Since, it has both roots equal
$\Rightarrow c^{2}-36=0\Rightarrow c=\pm6$
$\Rightarrow c=6\left\{m > 0 \Rightarrow c > 0\right\}$
$\Rightarrow m=1\&c=6$
$\Rightarrow \frac{c + m}{c - m}=\frac{7}{5}=1.4$