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

Let

A, A^{'}

be endpoints of major axis and

P

be any point on the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b)

. Prove that the locus of the intersection of

A P

with the straight line through

A^{'}

perpendicular to

A^{'} P

is a straight line which is perpendicular to the major axis.

Accepted Answer

Let the equation to the ellipse be $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ .

The co-ordinates of $A^{'}$ and $A$ will be $(a, 0)$ and $A^{'}$ , respectively.

If $P (a \cos ϕ, b \sin ϕ)$ be any point on the ellipse, then the equation to $A P$ will be,

$y - y_{1} = m (x - x_{1})$

By putting values, $y - 0 = \frac{b \sin ϕ - 0}{a \cos ϕ + a} (x + a)$

or $y = \frac{b}{a} \frac{2 \sin \frac{ϕ}{2} \cos \frac{ϕ}{2}}{2 \cos^{2} \frac{ϕ}{2}} (x + a)$

or $y = \frac{b}{a} \tan \frac{ϕ}{2} (x + a) . \dots (1)$

Similarly, equation of $A^{'} P$ is $y - y_{1} = m (x - x_{1})$ , which gives $y - 0 = \frac{b \sin ϕ - 0}{a \cos ϕ - a} (x - a)$

Or $y = - \frac{b}{a} \cot (\frac{ϕ}{2}) (x - a) (∵ 1 - \cos ϕ = 2 \sin^{2} \frac{ϕ}{2})$ .

Equation of line through $A^{'}$ and perpendicular to $A^{'} P$ ,

$y = (\frac{a}{b}) \tan (\frac{ϕ}{2}) (x - a) . \dots (2)$

The required locus of the point of intersection of $(1)$ and $(2)$ will be obtained by eliminating the variable $ϕ$ from the two.

So dividing the equation $(1)$ by $(2)$ , we have

$1 = \frac{b^{2} (x + a)}{a^{2} (x - a)}$

It is clearly a straight line which is perpendicular to the $x$ -axis, i.e., the major axis.

Important Questions on The Ellipse

Learn and Prepare for any exam you want !

Let A, A' be endpoints of major axis and P be any point on the ellipse x2a2+y2b2=1a>b. Prove that the locus of the intersection of AP with the straight line through A'perpendicular to A'P is a straight line which is perpendicular to the major axis.

Important Questions on The Ellipse

Learn and Prepare for any exam you want !