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

If

α, β, γ

and

δ

be the eccentric angles of the four points of intersection of the ellipse and any circle, prove that

α + β + γ + δ

is an even multiple of

π

radians.

Accepted Answer

Let the equation of the ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and the equation of the circle is $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
Suppose one of the points of intersection of the circle and the ellipse has eccentric angle $θ$

Then the point $(a \cos θ, b \sin θ)$ will satisfy the equation to the circle.

$\Rightarrow a^{2} \cos^{2} θ + b^{2} \sin^{2} θ + 2 g a \cos θ + 2 f b \sin θ + c = 0$
Let $\tan \frac{θ}{2} = t$ $\Rightarrow \cos θ = \frac{1 - \tan^{2} \frac{θ}{2}}{1 + \tan^{2} \frac{θ}{2}} = \frac{1 - t^{2}}{1 + t^{2}}$ & $\sin θ = \frac{2 \tan \frac{θ}{2}}{1 + \tan^{2} \frac{θ}{2}} = \frac{2 t}{1 + t^{2}}$
$\Rightarrow a^{2} {(\frac{1 - t^{2}}{1 + t^{2}})}^{2} + b^{2} {(\frac{2 t}{1 + t^{2}})}^{2} + 2 g a (\frac{1 - t^{2}}{1 + t^{2}}) + 2 f b (\frac{2 t}{1 + t^{2}}) + c = 0$
$\Rightarrow a^{2} {(1 - t^{2})}^{2} + 4 b^{2} t^{2} + 2 g a (1 - t^{2}) (1 + t^{2}) + 2 f b 2 t \times (1 + t^{2}) + c {(1 + t^{2})}^{2} = 0$
$\Rightarrow t^{4} (a^{2} - 2 g a + c) + 4 f b t^{3} + t^{2} (4 b^{2} - 2 a^{2} + 2 c) + 4 f b t + (a^{2} + 2 g a + c) = 0$

Let the roots of the above equation are $t_{1}, t_{2}, t_{3}$ and $t_{4}$ , then
$s_{1} = \sum t_{1} = \frac{- 4 f b}{a^{2} - 2 g a + c}$

$s_{2} = \sum t_{1} t_{2} = \frac{4 b^{2} - 2 a^{2} + 2 c}{a^{2} - 2 g a + c}$
$s_{3} = \sum t_{1} t_{2} t_{3} = \frac{- 4 f b}{a^{2} - 2 g a + c}$

$s_{4} = t_{1} t_{2} t_{3} t_{4} = \frac{a^{2} + 2 g a + c}{a^{2} - 2 g a + c}$

Now, $\tan (\frac{θ_{1} + θ_{2} + θ_{3} + θ_{4}}{2}) = \frac{s_{1} - s_{3}}{1 - s_{2} + s_{4}} = 0$ ${∵ s_{1} = s_{3}}$
$\Rightarrow \frac{θ_{1} + θ_{2} + θ_{3} + θ_{4}}{2} = n π$
$\Rightarrow θ_{1} + θ_{2} + θ_{3} + θ_{4} = 2 n π$

If $α, β, γ$ and $δ$ be the eccentric angles of the four points of intersection of the ellipse and any circle, prove that $α + β + γ + δ$ is an even multiple of $π$ radians.

Important Questions on The Ellipse

Learn and Prepare for any exam you want !

If α,β,γ and δ be the eccentric angles of the four points of intersection of the ellipse and any circle, prove that α+β+γ+δ is an even multiple of π radians.

Important Questions on The Ellipse

Learn and Prepare for any exam you want !

If $α, β, γ$ and $δ$ be the eccentric angles of the four points of intersection of the ellipse and any circle, prove that $α + β + γ + δ$ is an even multiple of $π$ radians.