Line and the Ellipse - Practice Questions & MCQ

Line and the Ellipse

Equation of Ellipse and Line is

Ellipse : $\quad \frac{\mathrm{x}}{\mathrm{a}^2}+\frac{\mathrm{y}}{\mathrm{b}^2}=1$
Line: $\quad y=m x+c$
After solving Eq. (i) and Eq. (ii)

$
\begin{aligned}
& \frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{(\mathrm{mx}+\mathrm{c})^2}{\mathrm{~b}^2}=1 \\
& \Rightarrow \quad\left(\mathrm{a}^2 \mathrm{~m}^2+\mathrm{b}^2\right) \mathrm{x}^2+2 \mathrm{mca}^2 \mathrm{x}+\mathrm{c}^2 \mathrm{a}^2-\mathrm{a}^2 \mathrm{~b}^2=0
\end{aligned}
$
Above equation is quadratic in x

Now depending on the value of determinant of this equation we can have following cases

If $\mathrm{D}>0$, then we have 2 real and distinct roots which means two distinct points of intersection of the line and the ellipse.

If $D=0$, then we have 1 real and repeated root which means one point of intersection of the line and the ellipse which means that the line is tangent to the ellipse. By putting values in $\mathrm{D}=0$, we get $a^2 m^2+b^2=c^2$. This is also the condition of tangency for the line $y=m x+c$ to be tangent to the ellipse.

If $\mathrm{D}<0$, then we do not have any real root, which meansno point of intersection of the line and the ellipse.

Using $a^2 m^2+b^2=c^2$ as the condition of tangency for the line $y=m x+c$ to be tangent to the ellipse, the equation of tangent to the standard ellipse is

$y=m x \pm \sqrt{a^2 m^2+b^2}$