# Examples on Tangents and Chords of Hyperbolas Set 2

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Example - 24

From any point on the hyperbola  \begin{align}\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1,\end{align} tangents are drawn to the circle  $${{x}^{2}}+{{y}^{2}}={{c}^{2}}.$$ Prove that the chord of contact of these tangents touches the hyperbola  $${{a}^{2}}{{x}^{2}}-{{b}^{2}}{{y}^{2}}={{c}^{4}}.$$

Solution: Assume any point on the given hyperbola as $$P(a\sec \theta ,\,b\tan \theta ).$$ From P, tangents are drawn to the given circle. The equation of the chord of contact will be

\begin{align} &\qquad\;\; T(a\sec \theta ,\,b\tan \theta )=0 \\ & \Rightarrow \quad ax\sec \theta +by\tan \theta ={{c}^{2}} \\\\ & \Rightarrow\quad \frac{x}{{}^{{{c}^{2}}}\!\!\diagup\!\!{}_{a}\;}\sec \theta +\frac{y}{{}^{{{c}^{2}}}\!\!\diagup\!\!{}_{b}\;}\tan \theta =1 \\\\ & \Rightarrow\quad \frac{x}{{}^{{{c}^{2}}}\!\!\diagup\!\!{}_{a}\;}\sec \phi -\frac{y}{{}^{{{c}^{2}}}\!\!\diagup\!\!{}_{b}\;}\tan \phi =1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\phi =-\theta ) \\\\ \end{align}

This is evidently a tangent to the hyperbola

\begin{align}&\qquad \frac{{{x}^{2}}}{{{A}^{2}}}-\frac{{{y}^{2}}}{{{B}^{2}}}=1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( A=\frac{{{c}^{2}}}{a},\,B=\frac{{{c}^{2}}}{b} \right) \\ & \Rightarrow\quad {{a}^{2}}{{x}^{2}}-{{b}^{2}}{{y}^{2}}={{c}^{4}} \\ \end{align}

Example - 25

From a point P, tangents are drawn to the hyperbola  \begin{align}\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1.\end{align}  The chord of contact of these tangents subtends a right angle at the centre C of the hyperbola. Determine the locus of P.

Solution: Let the point P be $$(h,\,k).$$

The equation of the chord of contact (say QR) of the tangents through P is

\begin{align} &\qquad\;\; T(h,\,k)=0 \\ & \Rightarrow\quad \frac{hx}{{{a}^{2}}}-\frac{ky}{{{b}^{2}}}=1 \\\end{align}

QR subtends a right angle at the origin.

We first evaluate the joint equation of CQ and CR by homogenizing the equation of the hyperbola using the equation of QR :

$\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}={{\left( \frac{hx}{{{a}^{2}}}-\frac{ky}{{{b}^{2}}} \right)}^{2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ldots \left( 1 \right)$

Since $$CQ\bot CR,$$  in (1), we must have

\begin{align}&\qquad\;\; \text{Coeff}\text{. of }{{x}^{2}}+\text{Coeff}\text{. of }{{y}^{2}}=0 \\ & \Rightarrow\quad \frac{1}{{{a}^{2}}}-\frac{{{h}^{2}}}{{{a}^{4}}}-\frac{1}{{{b}^{2}}}-\frac{{{k}^{2}}}{{{b}^{4}}}=0 \\ & \Rightarrow\quad \frac{{{h}^{2}}}{{{a}^{4}}}+\frac{{{k}^{2}}}{{{b}^{4}}}=\frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}} \\\end{align}

Thus, the locus of P is evidently an ellipse with the equation

$\frac{{{x}^{2}}}{{{a}^{4}}}+\frac{{{y}^{2}}}{{{b}^{4}}}=\frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}}$

## TRY YOURSELF - IV

Q 1. Find the point of intersection of the tangents drawn to the hyperbola  \begin{align}\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\end{align} at the points where it is intersected by the line $$px+qy+r=0.$$

Q 2. From any point P on the auxiliary circle of the hyperbola  $${{x}^{2}}-{{y}^{2}}={{a}^{2}},$$  tangents PQ and PR are drawn to the hyperbola. Find the locus of the mid-point of QR.

Q 3. Chords of the hyperbola $${{x}^{2}}-{{y}^{2}}={{a}^{2}}$$ touch the parabola  $${{y}^{2}}=4ax.$$ Prove that the locus of their mid-points is  $${{y}^{2}}(a-x)={{x}^{3}}.$$

Hyperbolas
grade 11 | Questions Set 1
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grade 11 | Questions Set 2
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