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  \({{y}^{2}}(a-x)={{x}^{3}}.\)