Line and a Parabola - Practice Questions & MCQ

Line and a Parabola

To get the point(s) of intersection, let us solve the equations of the parabola and the line simultaneously
Parabola is $\mathrm{y}^2=4 a x$ and a line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$
then, $\mathrm{y}^2=4 \mathrm{a}\left(\frac{\mathrm{y}-\mathrm{c}}{\mathrm{m}}\right)$

$
\Rightarrow \mathrm{my}^2-4 \mathrm{ay}+4 \mathrm{ac}=0
$
The above equation is quadratic in $y$
Depending on the discriminant of this equation, if we have 2 real roots, then 2 distinct intersection points and line is a chord/secant If we have 2 equal roots, then we have only one point where line touches the parabola and the line is a tangent
If we do not have any real roots, then line does not intersect the parabola
Condition of tangency: The line $y=m x+c$ will be $a$ tangent to the parabola $y^2=4 a x$, if $D=0 \Rightarrow c=a / m$

The point of contact :
Substitute the value of $c=a / m$ in the equation $m^2-4 a y+4 a c=0$

$
\begin{array}{rlrl} 
& & \mathrm{my}^2-4 \mathrm{ay}+4 \mathrm{a}\left(\frac{\mathrm{a}}{\mathrm{~m}}\right) & =0 \\
\Rightarrow & \mathrm{~m}^2 \mathrm{y}^2-4 \mathrm{amy}+4 \mathrm{a}^2 & =0 \\
\Rightarrow & (\mathrm{my}-2 \mathrm{a})^2 & =0 \\
\Rightarrow & \mathrm{my}-2 \mathrm{a} & =0 \quad \text { or } \quad \mathrm{y}=\frac{2 \mathrm{a}}{\mathrm{~m}}
\end{array}
$

Now, substitute the value of ' $y^{\prime}$ in $y=m x+\frac{a^m}{m}$ wre get, $x=\frac{a}{\mathrm{~m}^2}$