Question

The eccentricity of the hyperbola whose latus rectum is $$8$$ and conjugate axis is equal to half the distance between the foci is :

A. $$\frac{4}{3}$$
B. $$\frac{4}{{\sqrt 3 }}$$
C. $$\frac{2}{{\sqrt 3 }}$$  
D. none of these
Answer :   $$\frac{2}{{\sqrt 3 }}$$
Solution :
The standard equation of hyperbola is $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$$
Latus rectum $$ = \frac{{2{b^2}}}{a},$$
Conjugate axis $$ = 2b,$$
Distance between the foci $$= 2 ae$$
According to the question,
$$\eqalign{ & \frac{{2{b^2}}}{a} = 8......\left( {\text{i}} \right) \cr & 2b = \frac{1}{2}\left( {2ae} \right) \Rightarrow b = \frac{{ae}}{2}......\left( {{\text{ii}}} \right) \cr} $$
From equation $$\left( {\text{i}} \right)\& \left( {{\text{ii}}} \right),$$
$$\eqalign{ & \frac{2}{a}{\left( {\frac{{ae}}{2}} \right)^2} = 8 \cr & \Rightarrow 2.\frac{{{a^2}{e^2}}}{{4a}} = 8 \cr & \Rightarrow a{e^2} = 16......\left( {{\text{iii}}} \right) \cr} $$
From equation $$\left( {\text{i}} \right),\,\,{b^2} = 4a$$
Using $${b^2} = {a^2}\left( {{e^2} - 1} \right)$$    we get
$$\eqalign{ & \left( {4a} \right) = {a^2}\left( {{e^2} - 1} \right) \cr & \Rightarrow 4 = \frac{{16}}{{{e^2}}}\left( {{e^2} - 1} \right) \cr & \Rightarrow 16 - \frac{{16}}{{{e^2}}} = 4 \cr & \Rightarrow \frac{{16}}{{{e^2}}} = 12 \cr & \therefore \,e = \frac{2}{{\sqrt 3 }}. \cr} $$

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. $${y^2} - {x^2} \leqslant 0$$
View Answer
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. $$ - \left( {\frac{{{a^2} + {b^2}}}{b}} \right)$$
View Answer
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. $$9{x^2} - 8{y^2} + 18x + 9 = 0$$
View Answer
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
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