Question

If in a hyperbola the eccentricity is $$\sqrt 3 ,$$  and the distance between the foci is 9 then the equation of the hyperbola in the standard form is :

A. $$\frac{{{x^2}}}{{{{\left( {\frac{{\sqrt 3 }}{2}} \right)}^2}}} - \frac{{{y^2}}}{{{{\left( {\sqrt {\frac{3}{2}} } \right)}^2}}} = 1$$
B. $$\frac{{{x^2}}}{{{{\left( {\frac{{3\sqrt 3 }}{2}} \right)}^2}}} - \frac{{{y^2}}}{{{{\left( {\frac{{3\sqrt 3 }}{2}} \right)}^2}}} = 1$$  
C. $$\frac{{{x^2}}}{{{{\left( {\frac{{3\sqrt 3 }}{2}} \right)}^2}}} - \frac{{{y^2}}}{{{{\left( {\frac{{3\sqrt 2 }}{2}} \right)}^2}}} = 1$$
D. none of these
Answer :   $$\frac{{{x^2}}}{{{{\left( {\frac{{3\sqrt 3 }}{2}} \right)}^2}}} - \frac{{{y^2}}}{{{{\left( {\frac{{3\sqrt 3 }}{2}} \right)}^2}}} = 1$$
Solution :
For rectangular hyperbola, $$e = \sqrt 2 .$$  So, $${\left( {x - 1} \right)^2} + {\left( {y - 2} \right)^2} = {\left( {\sqrt 2 .\frac{{x + y - 1}}{{\sqrt 2 }}} \right)^2}.$$

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
View Answer

Practice More Releted MCQ Question on
Hyperbola

Practice More MCQ Question on Maths Section

AlgebraSequences and SeriesQuadratic EquationComplex NumberMatrices and DeterminantsPermutation and CombinationBinomial TheoremMathematical InductionMathematical Reasoning
Statistics and ProbabilityStatisticsProbability
CalculusSets and RelationsFunctionLimitsContinuityDifferentiability and DifferentiationApplication of DerivativesDifferential EquationsIndefinite IntegrationDefinite IntegrationApplication of Integration
GeometryStraight LinesCircleParabolaEllipseHyperbolaLocus3D Geometry and VectorsThree Dimensional Geometry
TrigonometryTrigonometric Ratio and IdentitiesTrignometric EquationsProperties and Solutons of TriangleInverse Trigonometry Function