Examples on Tangents and Normals To Rectangular Hyperbolas Set 2
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 \(FP \cdot F'P = C{P^2}.\)