Name: Equation of Chord of Hyperbola in Parametric Form
Uploaded: 2019-07-19
Duration: 1 min 30 s
Description: How to find the equation of chord of a hyperbola joining two points whose parametric coordinates are known? Check out this video to find out.

Question

Prove that the locus of the middle point of the chord of contact of tangents from any point of the circle

x^{2} + y^{2} = r^{2}

to the hyperbola

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

is given by the equation

{(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}})}^{2} = (\frac{x^{2} + y^{2}}{r^{2}}) .

Accepted Answer

Let, $(r \cos θ, r \sin θ)$ be the point on the circle $x^{2} + y^{2} = r^{2} .$

The equation of chord of the contact of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ drawn from the point $(r \cos θ, r \sin θ)$ is $\frac{x \cdot r \cos θ}{a^{2}} - \frac{y \cdot r \sin θ}{b^{2}} = 1$ .

$\Rightarrow \frac{x \cdot \cos θ}{a^{2}} - \frac{y \cdot \sin θ}{b^{2}} = \frac{1}{r} . . . . . (i)$

Let, $(h, k)$ be the mid-point of chord of contact of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 .$

Then equation of chord of contact is $\frac{h x}{a^{2}} - \frac{k y}{b^{2}} = \frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}} . . . (ii)$

On comparing $(i) & (ii)$

$\Rightarrow \frac{\frac{\cos θ}{a^{2}}}{\frac{h}{a^{2}}} = \frac{- \frac{\sin θ}{b^{2}}}{- \frac{k}{b^{2}}} = \frac{\frac{1}{r}}{\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}}}$

$\Rightarrow \frac{\cos θ}{h} = \frac{\sin θ}{k} = \frac{1}{r (\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}})}$

$\Rightarrow \cos θ = \frac{h}{r (\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}})} & \sin θ = \frac{k}{r (\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}})}$

$\Rightarrow \cos^{2} θ + \sin^{2} θ = \frac{h^{2}}{r^{2} {(\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}})}^{2}} + \frac{k^{2}}{r^{2} {(\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}})}^{2}}$

$\Rightarrow 1 = \frac{h^{2} + k^{2}}{r^{2} {(\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}})}^{2}}$

$\Rightarrow {(\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}})}^{2} = \frac{h^{2} + k^{2}}{r^{2}}$

Replace $h$ by $x$ and $k$ by $y$

$\Rightarrow {(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}})}^{2} = \frac{x^{2} + y^{2}}{r^{2}}$

Important Questions on Hyperbola

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Prove that the locus of the middle point of the chord of contact of tangents from any point of the circle x2+y2=r2 to the hyperbola x2a2-y2b2=1 is given by the equation x2a2-y2b22=x2+y2r2.

Important Questions on Hyperbola

Learn and Prepare for any exam you want !