Examples on Tangents to Hyperbolas Set 2
Example - 14
Chords of the circle \({x^2} + {y^2} = {a^2}\) touch the hyperbola \(\begin{align}\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1.\end{align}\) Find the locus of the mid-points of such chords.
Solution : Assume the coordinates of the mid-point P as (h, k); we wish to determine the locus of P.
A chord of the circle \({x^2} + {y^2} = {a^2}\) bisected at the point P has the equation
\[\begin{align}&\qquad \quad \;\;\;\;T(h,\,\,k) = S(h,\,\,k) \\\\ &\Rightarrow \qquad hx + ky - {a^2} = {h^2} + {k^2} - {a^2} \\\\&\Rightarrow \qquad y = \left( { - \frac{h}{k}} \right)x + \left( {\frac{{{h^2} + {k^2}}}{k}} \right) \\ \end{align} \]
This is a tangent to the given hyperbola if the condition for tangency \(({c^2} = {a^2}{m^2} - {b^2})\) is satisfied, i.e. if
\[\begin{align}&\qquad\;\;\;\;\;{\left( {\frac{{{h^2} + {k^2}}}{k}} \right)^2} = {a^2}{\left( { - \frac{h}{k}} \right)^2} - {b^2} \\\\&\Rightarrow \qquad {({h^2} + {k^2})^2} = {a^2}{h^2} - {b^2}{k^2}\\\end{align} \]
Thus, we see that the locus of the mid point is
\[{({x^2} + {y^2})^2} = {a^2}{x^2} - {b^2}{y^2}\]
Example - 15
From the center O of the hyperbola \(\begin{align}\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1,\end{align}\) perpendicular OP is dropped upon any tangent to the hyperbola. Find the locus of P.
Solution :
Assume the coordinates of P to be (h, k).
Any tangent to the given hyperbola has the form
\[y = mx + \sqrt {{a^2}{m^2} - {b^2}} \]
Since P lies on this tangent, we have
\[k = mh + \sqrt {{a^2}{m^2} - {b^2}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \ldots \left( 1 \right)\]
Also, since OP is perpendicular to this tangent, we have
\[\begin{align}&\qquad\;\;\frac{k}{h} \times m = - 1 \\ &\Rightarrow \quad m = - \frac{h}{k}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \ldots \left( 2 \right)\\\end{align} \]
Using (2) in (1), we obtain a relation in h and k:
\[{({h^2} + {k^2})^2} = {a^2}{h^2} - {b^2}{k^2}\]
Thus, the locus of P is
\[{({x^2} + {y^2})^2} = {a^2}{x^2} - {b^2}{y^2}\]
TRY YOURSELF - II
Q1. (a) Prove that the locus of the point of intersection of two perpendicular tangents to a hyperbola \(\begin{align}\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\end{align}\) will be the circle
\[{x^2} + {y^2} = {a^2} - {b^2}\]
As for the other conics, such a circle is termed the Director circle of the hyperbola.
(b). Generalize the result of part (a). Prove that the locus of the point of intersection of two tangents to \(\begin{align}\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\end{align}\) inclined at an angle \(\theta \) will be
\[{({x^2} + {y^2} + {b^2} - {a^2})^2} = 4({a^2}{y^2} - {b^2}{x^2} + {a^2}{b^2}){\cot ^2}\theta \]
Q2. Prove that the locus of the mid-point of the segment of a tangent to \(\begin{align}\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}\end{align}\) intercepted between the coordinate axes is
\[{a^2}{y^2} - {b^2}{x^2} = 4{x^2}{y^2}.\]
Q3. Find the condition for the line \(x\cos \alpha + y\sin \alpha = p\) to be a tangent to the hyperbola \(\begin{align}\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\end{align}\).
Q4. Find the equations of the tangents to the hyperbola \(3{x^2} - {y^2} = 3\) which are perpendicular to \(x + 3y = 2.\)
Q5. Find the equation(s) of the common tangents to the parabola \({y^2} = 8x\) and the hyperbola \(3{x^2} - {y^2} = 3.\)