Name: Conic Sections and Ellipse
Uploaded: 2019-07-19
Duration: 3 min 55 s
Description: In this video, we find the equation of an ellipse that is centred at the origin when the eccentricity and vertices are given. Study this video for a better understanding.

Question

P Q

is a focal chord of an ellipse. The tangents at

P

and

Q

intersect in

T

and the normals at

P, Q

intersect in

R

. Show that

T R

passes through the other focus.

Accepted Answer

Let th equation ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ .

Let $P$ is $(a \cos α, b \sin α)$ and $Q$ is $(a \cos β, b \sin β)$ , then tangent at $P$ and $Q$ intersect at $T$ , so

$T \equiv (\frac{a \cos (\frac{α + β}{2})}{\cos (\frac{α - β}{2})}, \frac{b \sin (\frac{α + β}{2})}{\cos (\frac{α - β}{2})})$

Normal at $P$ and $Q$ interest at $R$ , then $R$ is

$R \equiv ((\frac{a^{2} - b^{2}}{a}) \frac{\cos a \cos b \cdot \cos (\frac{α + β}{2})}{\cos (\frac{α - β}{2})}, \frac{- (a^{2} - b^{2})}{a} \frac{\sin a \sin b \sin (\frac{α + β}{2})}{\cos (\frac{α - β}{2})})$

Equation of $P Q$ is

$\frac{x}{a} \cos (\frac{α + β}{2}) + \frac{y}{b} \sin (\frac{α + β}{2}) - \cos (\frac{α - β}{2}) = 0$

$P Q$ is focal chord also so must be passes through $(a e, 0)$ , so

$e \cos (\frac{α + β}{2}) = \cos (\frac{α - β}{2})$

$\Rightarrow e = \frac{\cos (\frac{α - β}{2})}{\cos (\frac{α + β}{2})} . . . . (1)$

Equation of $T R$ is

$y - (\frac{b \sin (\frac{α + β}{2})}{\cos (\frac{α - β}{2})}) = [\frac{(\frac{- (a^{2} - b^{2})}{a} \frac{\sin a \sin b \sin (\frac{α + β}{2})}{\cos (\frac{α - β}{2})} - \frac{b \sin (\frac{α + β}{2})}{\cos (\frac{α - β}{2})})}{(\frac{a^{2} - b^{2}}{a}) \frac{\cos a \cos b \cdot \cos (\frac{α + β}{2})}{\cos (\frac{α - β}{2})} - \frac{a \cos (\frac{α + β}{2})}{\cos (\frac{α - β}{2})}}] (x - \frac{a \cos (\frac{α + β}{2})}{\cos (\frac{α - β}{2})})$

Using $(1)$ and putting $(- a e, 0)$ (another focus) which must satisfy above equation.

$P Q$ is a focal chord of an ellipse. The tangents at $P$ and $Q$ intersect in $T$ and the normals at $P, Q$ intersect in $R$ . Show that $T R$ passes through the other focus.

Important Questions on Conic Section

Learn and Prepare for any exam you want !

PQ is a focal chord of an ellipse. The tangents at P and Q intersect in T and the normals at P,Q intersect in R. Show that TR passes through the other focus.

Important Questions on Conic Section

Learn and Prepare for any exam you want !

$P Q$ is a focal chord of an ellipse. The tangents at $P$ and $Q$ intersect in $T$ and the normals at $P, Q$ intersect in $R$ . Show that $T R$ passes through the other focus.