the foci of the ellipse x 2a 2 y 2b 2 1 and the hyperbola x

The foci of the ellipse $$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ and the hyperbola $$\frac{{{x^2}}}{{144}} - \frac{{{y^2}}}{{81}} = \frac{1}{{25}}$$ coincide. Then the value of $${b^2}$$ is :

A. 5

B. 7

C. 9

D. 1

Answer : 7

Solution :
For the ellipse, $${a^2} = 16;\,\,{b^2} = {a^2}\left( {1 - {e^2}} \right)$$
$$ \Rightarrow e = \frac{{\sqrt {16 - {b^2}} }}{4}\,\,\,\, \Rightarrow ae = \sqrt {{{16}^2} - {b^2}} $$
For the hyperbola, $${a^2} = \frac{{144}}{{25}},\,\,{b^2} = \frac{{81}}{{25}};\,\,{b^2} = {a^2}\left( {{e^2} - 1} \right) \Rightarrow e = \frac{5}{4} \Rightarrow ae = 3$$
$$\therefore \,\,\,\sqrt {16 - {b^2}} = 3$$

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

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

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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

View Answer

The foci of the ellipse $$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ and the hyperbola $$\frac{{{x^2}}}{{144}} - \frac{{{y^2}}}{{81}} = \frac{1}{{25}}$$ coincide. Then the value of $${b^2}$$ is :

Releted MCQ Question on
Geometry >> Hyperbola

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 :

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

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Hyperbola

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The foci of the ellipse $$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ and the hyperbola $$\frac{{{x^2}}}{{144}} - \frac{{{y^2}}}{{81}} = \frac{1}{{25}}$$ coincide. Then the value of $${b^2}$$ is :

Releted MCQ Question on Geometry >> Hyperbola

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 :

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

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Releted MCQ Question on
Geometry >> Hyperbola

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Hyperbola