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Examples on Tangents and Normals To Rectangular Hyperbolas Set 1

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Example - 31

Show that, in general, four normals can be drawn to the hyperbola \(xy = {c^2}\)  from any point  \(P(h,\,k)\) such that

(a) Sum of the x-coordinates of the feet of the normal = h

(b) Sum of the y-coordinates of the feet of the normal = k

(c) Product of the x-coordinates of the feet of the normal

= Product of the y-coordinates of the feet of the normal

=\( - {c^4}.\)

Solution: Any normal to  \(xy = {c^2}\) can be written in the form

\[x{t^3} - yt - c{t^4} + c = 0\]

If this passes through  \(P(h,\,k),\)  we have

\[\begin{align}&\qquad\;\; h{t^3} - kt - c{t^4} + c = 0 \\   & \Rightarrow\quad c{t^4} - h{t^3} + kt - c = 0 \\\end{align} \]

This biquadratic equation gives (in general) four roots for t corresponding to the four normals. Let the four roots be  \({t_1},\,{t_2},\,{t_3},\,{t_4}.\)

We have,

Sum of x-coordinates of feet of normals \(\sum\limits_{i = 1}^4 {c{t_i} = c\sum\limits_{i = 1}^4 {{t_i} = c \cdot \begin{align}\frac{h}{c}\end{align} = h} } \)
Sum of y-coordinates of feet of normals \(\begin{align}\sum\limits_{i = 1}^4 {\frac{c}{{{t_i}}} = c\sum\limits_{i = 1}^4 {\frac{1}{{{t_i}}} = c \cdot \frac{k}{c} = k} } \end{align}\)
Product of  x-coordinates of feet of normals \({c^4}\prod\limits_{i = 1}^4 {{t_i} =  - {c^4}} \)
Product of  y-coordinates of feet of normals  \({c^4}\prod\limits_{i = 1}^4 {\frac{1}{{{t_i}}} =  - {c^4}} \)

 This proves the assertions stated in the question.

Example - 32

A tangent is drawn to  \(xy = {c^2}\)  at the point P, intersecting the coordinates axes in A and B. Prove that area \((\Delta OAB)\)  is constant where O is the origin.

Solution: Although we’ve already discussed the general problem(of which this problem is a particular example) in Example 29, we redo it here just to gain practice with tangents to \(xy = {c^2}.\)

Let the point P be  \(\left( {ct,\,\begin{align}\frac{c}{t}\end{align}} \right).\) The tangent at P has the equation

\[x + y{t^2} = 2ct\]

This intersects the x-axis in

\[A \equiv (2ct,\,0)\]

and the y - axis in

\[B \equiv \left( {0,\,\frac{{2c}}{t}} \right)\]

Thus, area \((\Delta OAB) =\begin{align} \frac{1}{2} \times 2ct \times \frac{{2c}}{t}\end{align}\)

 \( = 2{c^2}\) (a constant)

Also notice that, as in Example -29, P is the mid-point of AB.

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Download SOLVED Practice Questions of Examples on Tangents and Normals To Rectangular Hyperbolas Set 1 for FREE
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