Name: Pair of Tangent, Chord of Contact
Uploaded: 2019-07-19
Duration: 19 min 37 s
Description: This video contains all the details of the chord of the contact pair of tangent and cord bisected at a point. This concept is very useful for the students.

Question

Any point

P

of an ellipse is joined to the extremities of the major axis; prove that the portion of a directrix intercepted by them subtends a right angle at the corresponding focus

Accepted Answer

Let the equation of ellipse be given as,

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

And, let any point $P$ on ellipse be $(a \cos θ, b \sin θ)$ and $A$ and $A'$ be the end points of the major axis.

Then, the equation of the line $P A$ is given as,

$y - 0 = \frac{b \sin θ - 0}{a \cos θ - a} (x - a)$

The coordinates of the point of the intersection $N$ on the line $P A$ with the directrix $x = \frac{a}{e}$ are $(\frac{a}{e}, b \sin θ \frac{(1 - e)}{e (\cos θ - 1)})$ .

Similarly, the equation of the line $P A^{'}$ is given as,

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

The line $P A'$ and the directrix $x = \frac{a}{e}$ meet at point $M$ whose coordinates are $(\frac{a}{e}, b \sin θ \frac{(1 + e)}{e (\cos θ + 1)})$ .

Now $S (a e, 0)$ & we use $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Now, the product of the gradients of the lines $S N$ and $S M$ can be given as,

$\Rightarrow (\frac{\frac{b \sin θ (1 - e)}{e (\cos θ - 1)}}{\frac{a}{e} - a e}) (\frac{\frac{b \sin θ (1 + e)}{e (\cos θ + 1)}}{\frac{a}{e} - a e})$

$= (\frac{\frac{b sinθ (1 - e)}{e (cosθ - 1)} \times e}{a (1 - e^{2})}) (\frac{\frac{b sinθ (1 + e)}{e (cosθ + 1)} \times e}{a (1 - e^{2})})$

$= \frac{b^{2} (1 - e^{2}) \times \sin^{2} θ}{(\cos^{2} θ - 1) \times a^{2} {(1 - e^{2})}^{2}}$

$= \frac{a^{2} (1 - e^{2}) (1 - e^{2}) \sin^{2} θ}{- a^{2} {(1 - e^{2})}^{2} (1 - \cos^{2} θ)}$ ( $∵ b^{2} = a^{2} (1 - e^{2})$ )

$= - 1$

Hence, $S N$ is perpendicular to $S M$ . That is, $M N$ subtends a right angle at $S$ .

Any point $P$ of an ellipse is joined to the extremities of the major axis; prove that the portion of a directrix intercepted by them subtends a right angle at the corresponding focus

Important Questions on The Ellipse

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