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

Y

and

Z

are the feet of the perpendiculars from the foci upon the tangent at any point

P

of an ellipse

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

, prove that the tangents at

Y

and

Z

to the auxiliary circle, meet on the ordinates of

P

, and that the locus of their point of intersection is another ellipse.

Accepted Answer

Taking the coordinates of $P$ on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ as $(a \cos ϕ, b \sin ϕ)$ , we get the equation of the tangent at this point as,

$\frac{x}{a} \cos ϕ + \frac{y}{b} \sin ϕ = 1 \dots (1)$

The equation of the auxiliary circle can be given as,

$x^{2} + y^{2} = a^{2}$

Let the feet of the perpendiculars from the foci on the tangent at $P$ be $Y$ and $Z$ .

Let $(h, k)$ be the coordinates of the point of intersection of the tangents on $Y$ and $Z$ . Then, $Y Z$ will become the chord of contact of $(h, k)$ . So, the equation of $Y Z$ is given as,

$x h + y k = a^{2} \dots (2)$

Equations $(1)$ and $(2)$ must be identical. So, comparing them, we get,

$h = a \cos ϕ$

$k = (\frac{a^{2}}{b}) \sin ϕ$

Hence, the ordinate of $P$ lies on $x = a \cos ϕ$ .

To get the locus of $(h, k)$ , we have to eliminate $ϕ$ .

We have,

$\cos ϕ = \frac{h}{a}$

$\sin ϕ = \frac{b k}{a^{2}}$

Hence, on squaring and adding the equations, we get,

$1 = \frac{h^{2}}{a^{2}} + \frac{b^{2} k^{2}}{a^{4}}$

$a^{2} h^{2} + b^{2} k^{2} = a^{4}$

Generalizing the locus of $(h, k)$ , we get,

$a^{2} x^{2} + b^{2} y^{2} = a^{4}$ (which is an ellipse).

Important Questions on The Ellipse

Learn and Prepare for any exam you want !

If Y and Z are the feet of the perpendiculars from the foci upon the tangent at any point P of an ellipse x2a2+y2b2=1, prove that the tangents at Y and Z to the auxiliary circle, meet on the ordinates of P, and that the locus of their point of intersection is another ellipse.

Important Questions on The Ellipse

Learn and Prepare for any exam you want !