QUESTION

# Prove that the straight line $lx+my+n=0$ touches the parabola ${{y}^{2}}=4ax$ if $nl=a{{m}^{2}}$.

Hint: To prove the expression given in the question, write the general equation of the tangent to the parabola ${{y}^{2}}=4ax$ in the slope form. Compare the slope of this tangent with the equation of straight line given in the question. Simplify the terms to prove the expression given in the question.

We have to prove that the straight line $lx+my+n=0$ touches the parabola ${{y}^{2}}=4ax$ if $nl=a{{m}^{2}}$.
We will first write the general equation of the tangent to the parabola ${{y}^{2}}=4ax$ in the slope form.
We know that the equation of the tangent to the parabola having slope ‘c’ is $y=cx+\dfrac{a}{c}$.
Rearranging the terms of the above equation, we have ${{c}^{2}}x-cy+a=0.....\left( 1 \right)$.
We know that the equation of straight line $lx+my+n=0.....\left( 2 \right)$ is tangent to the parabola.
Comparing the coefficients of ‘x’, ‘y’, and constant of both the equations, we have $l={{c}^{2}}.....\left( 3 \right)$, $m=-c.....\left( 4 \right)$ and $n=a.....\left( 5 \right)$.
Multiplying equation (3) and (5), we have $nl=a{{c}^{2}}.....\left( 6 \right)$.
Squaring equation (4), we have ${{m}^{2}}={{c}^{2}}.....\left( 7 \right)$.
Substituting the value of equation (7) in equation (6), we have $nl=a{{m}^{2}}$.
Hence, we have proved that the line $lx+my+n=0$ is tangent to the parabola ${{y}^{2}}=4ax$ if $nl=a{{m}^{2}}$.