if x 9 is the chord of contact of the hyperbola x 2 y 2 9

If $$x = 9$$ is the chord of contact of the hyperbola $${x^2} - {y^2} = 9,$$ then the equation of the corresponding pair of tangents is :

A. $$9{x^2} - 8{y^2} + 18x - 9 = 0$$

B. $$9{x^2} - 8{y^2} - 18x + 9 = 0$$

C. $$9{x^2} - 8{y^2} - 18x - 9 = 0$$

D. $$9{x^2} - 8{y^2} + 18x + 9 = 0$$

Answer : $$9{x^2} - 8{y^2} - 18x + 9 = 0$$

Solution :
The equation of tangent at a point $$\left( {{x_1},\,{y_1}} \right)$$ on the hyperbola $${x^2} - {y^2} = c$$ is given by $$x{x_1} - y{y_1} = c$$
Chord $$x = 9$$ meets $${x^2} - {y^2} = 9$$ at $$\left( {9,\,6\sqrt 2 } \right)$$ and $$\left( {9,\, - 6\sqrt 2 } \right)$$ at which tangents are
$$\eqalign{ & 9x - 6\sqrt 2 y = 9{\text{ and }}9x + 6\sqrt 2 y = 9 \cr & {\text{or, }}3x - 2\sqrt 2 y - 3 = 0{\text{ and }}3x + 2\sqrt 2 y - 3 = 0 \cr} $$
$$\therefore $$ Combined equation of tangents is
$$\eqalign{ & \left( {3x - 2\sqrt 2 y - 3} \right)\left( {3x + 2\sqrt 2 y - 3} \right) = 0 \cr & {\text{or, }}9{x^2} - 8{y^2} - 18x + 9 = 0 \cr} $$

Releted MCQ Question on
Geometry >> Hyperbola

Releted Question 1

Each of the four inequalities given below defines a region in the $$xy$$ plane. One of these four regions does not have the following property. For any two points $$\left( {{x_1},\,{y_1}} \right)$$ and $$\left( {{x_2},\,{y_2}} \right)$$ in the the region, the point $$\left( {\frac{{{x_1} + {x_2}}}{2},\,\frac{{{y_1} + {y_2}}}{2}} \right)$$ is also in the region. The inequality defining this region is :

A. $${x^2} + 2{y^2} \leqslant 1$$

B. $${\text{max }}\left\{ {\left| x \right|,\left| y \right|} \right\} \leqslant 1$$

C. $${x^2} - {y^2} \leqslant 1$$

D. $${y^2} - {x^2} \leqslant 0$$

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Releted Question 2

Let $$P\left( {a\,\sec \,\theta ,\,b\,\tan \,\theta } \right)$$ and $$Q\left( {a\,\sec \,\phi ,\,b\,\tan \,\phi } \right),$$ where $$\theta + \phi = \frac{\pi }{2},$$ be two points on the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1.$$ If $$\left( {h,\,k} \right)$$ is the point of intersection of the normal at $$P$$ and $$Q,$$ then $$k$$ is equal to :

A. $$\frac{{{a^2} + {b^2}}}{a}$$

B. $$ - \left( {\frac{{{a^2} + {b^2}}}{a}} \right)$$

C. $$\frac{{{a^2} + {b^2}}}{b}$$

D. $$ - \left( {\frac{{{a^2} + {b^2}}}{b}} \right)$$

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Releted Question 3