Question

If any point on a hyperbola has the coordinates $$\left( {5\tan \,\phi ,\,4\sec \,\phi } \right)$$ then the eccentricity of the hyperbola is :

A. $$\frac{5}{4}$$

B. $$\frac{{\sqrt {41} }}{5}$$

C. $$\frac{{25}}{{16}}$$

D. $$\frac{{\sqrt {41} }}{4}$$

Answer : $$\frac{{\sqrt {41} }}{4}$$

Solution :
$$\eqalign{ & x = 5\tan \,\phi ,\,\,y = \,4\sec \,\phi \, \Rightarrow \frac{{{y^2}}}{{16}} - \frac{{{x^2}}}{{25}} = 1\,\, \Rightarrow {a^2} = 16,\,{b^2} = 25 \cr & {\text{Also, }}{b^2} = {a^2}\left( {{e^2} - 1} \right) \cr} $$

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

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

Releted Question 3

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

Releted Question 4

For hyperbola $$\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1,$$ which of the following remains constant with change in $$'\alpha \,'$$

A. abscissae of vertices

B. abscissae of foci

C. eccentricity

D. directrix

Practice More Releted MCQ Question on
Hyperbola