Find the condition for $y=mx+c$ to cut $y^2=4ax$ at two distinct points, where $a>0$.

Question

Find the condition for the line $y=mx+c$ to cut the curve $y^2=4ax$ at two distinct points, where $a, m, c$ are constants and $a>0$.

Answer 1

Look at the quadratic equation $(mx+c)^2=4ax$ .

We get two distinct points $ \iff$ the discriminant $ \Delta$ of this quadratic equation is $>0.$

Can you proceed ?

Answer 2

Put $mx + c$ in place of $y$ in the parabola eqn, you will get a quadratic in $x$. Then apply $b^2 - 4ac > 0$ which is condition for 2 real solution in a quadratic.

You will get the condition: ($a>0$ and $mc<1$) or ($a<0$ and $mc>1$)