Examples on Tangents and Normals To Rectangular Hyperbolas Set 2

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Example - 33

Suppose that a normal drawn to the hyperbola \(xy = {c^2}\) at the point  \({t_1},\) meets it again at \({t_2}.\) Show that

\[{t_2} =  - \frac{1}{{t_1^3}}\]

Solution: The equation of the normal at  \({t_1}\) is

\[xt_1^3 - y{t_1} - ct_1^4 + c = 0\]

If this meets the hyperbola again at \({t_2}\left( {c{t_2},\,\begin{align}\frac{c}{{{t_2}}}\end{align}} \right),\)  we have

\[\begin{align}&\qquad\;\; c{t_2}t_1^3 - \frac{{c{t_1}}}{{{t_2}}} - ct_1^4 + c = 0 \\   &\Rightarrow\quad  t_2^2t_1^3 - t_1^4{t_2} + {t_2} - {t_1} = 0 \\\\   &\Rightarrow\quad  (t_1^3{t_2} + 1)({t_1} - {t_2}) = 0  \\ \end{align} \]

Since \({t_1} \ne {t_2},\)  we have

\[\begin{align}&\qquad\;\; t_1^3{t_2} + 1 = 0 \\   &\Rightarrow\quad   \boxed{{t_2} =  - \frac{1}{{t_1^3}}}  \\ \end{align} \]

Example - 34

The normal at a point  \({A_1}({t_1})\)  on the hyperbola  \(xy = {c^2}\)  meets the curve again at the \({A_2};\) normal at  \({A_2}\)  meets the curve again at \({A_3}\) and so on. Find the coordinates of   \({A_n}\) in terms of \({t_1}.\)

Solution: Let the point  \({A_n}\) be \({t_n}\left( {c{t_n},\,\begin{align}\frac{c}{{{t_n}}}\end{align}} \right).\)

Using the result of the last example, we have

\[\begin{align}&{t_2} =  - \frac{1}{{t_1^3}}  \\  &{t_3} =  - \frac{1}{{t_2^3}} = t_1^9  \\  &{t_4} =  - \frac{1}{{t_3^3}} =  - \frac{1}{{t_1^{27}}}  \\  &{t_n} = {( - 1)^{n - 1}}{t_1}^{{{( - 3)}^{n - 1}}}  \\\end{align} \]

It would be a good exercise for you to start with a particular value of   \({t_1}\) and see what points you obtain on the hyperbola by drawing a sequence of normals as described in the question.

Example - 35

From a point P on  \(xy = {c^2}\)  tangents are drawn to \(xy = {a^2}\) . The chord of the contact intersects the asymptotes at A and B . If the origin is O, prove that area  \((\Delta OAB)\)  is constant.

Solution: Let P be the point  \(\left( {ct,\,\begin{align}\frac{c}{t}\end{align}} \right).\)  The chord AB is basically the chord of contact of tangents drawn from P to the hyperbola  \(xy = {a^2};\)  thus, the equation of AB is

\[\begin{align}&\qquad\;\;\; T\left( {ct,\,\frac{c}{t}} \right) = 0 \\   &\Rightarrow\quad \frac{c}{t}x + cty = 2{a^2}  \\\end{align} \]

The points A and B can now easily be evaluated using this equation :

Point A                     Put         \(y = 0 \quad \Rightarrow \quad x = \begin{align}\frac{{2{a^2}t}}{c}\end{align}\)

\( \Rightarrow \quad A \equiv \left(\begin{align} {\frac{{2{a^2}t}}{c},\,0}\end{align} \right)\)

Point B                     Put         \(x = 0\quad \Rightarrow\quad y = \begin{align}\frac{{2{a^2}}}{{ct}}\end{align}\)

\( \Rightarrow\quad B \equiv \left( {0,\,\begin{align}\frac{{2{a^2}}}{{ct}} \end{align}}\right)\)

Thus,

area\((\Delta OAB) = \begin{align}\frac{1}{2}\end{align} \times OA \times OB\)

\(\begin{align} = \frac{1}{2} \times \frac{{2{a^2}t}}{c} \times \frac{{2{a^2}}}{{ct}}\end{align}\)

\( \begin{align}= \frac{{2{a^4}}}{{{c^2}}}\end{align}\)

The area is independent of t and hence is a constant.

TRY YOURSELF - V

Q. 1  Find the equation of the hyperbola whose asymptotes are  \(x + 2y + 3 = 0\) and  \(3x + 4y + 5 = 0\)  and which passes through (1, –1). What is the equation of its conjugate hyperbola ?

Q. 2  Find the equation of the asymptotes of the hyperbola  \(2{x^2} - 5xy - 3{y^2} - 5x - 3y - 21 = 0.\)

Q. 3  Find the locus of the mid-points of the chords of length 2a of the hyperbola  \(xy = {c^2}.\)

Q. 4  If F and F ' are the foci, C the centre and P any point on a rectangular hyperbola, show that \(FP \cdot F'P = C{P^2}.\)