# 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 $$FP \cdot F'P = C{P^2}.$$

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