Home
Class 12
MATHS
If hyperbola (x^2)/(b^2)-(y^2)/(a^2)=1 p...

If hyperbola `(x^2)/(b^2)-(y^2)/(a^2)=1` passes through the focus of ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` , then find the eccentricity of hyperbola.

Text Solution

Verified by Experts

Let `e_(1)` and `e_(2)` be the eccentricities of ellipse and hyperbola, respectively.

Since hyperbola passes through the foci of ellipse, we have
`b=ae_(1)" (1)"`
Also for ellipse, `b^(2)=a^(2)(1-e_(1)^(2))" (2)"`
and for hyperbola, `a^(2)=b^(2)(e_(2)^(2)-1)" (3)"`
From (1) and (2), we get
`2e_(1)^(2)=1 or e_(1)=(1)/(sqrt(2))`
So, from (1) and (3), we get
`2=e_(2)^(2)-1 or e_(2)=sqrt3`
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE ENGLISH|Exercise SOLVED EXAMPLES|11 Videos

  • HYPERBOLA

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 7.1|3 Videos

  • HIGHT AND DISTANCE

    CENGAGE ENGLISH|Exercise Archives|3 Videos

  • INDEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise Multiple Correct Answer Type|2 Videos

Similar Questions

Explore conceptually related problems

Find the eccentricity of hyperbola x^(2)-9y^(2)=1 .

If the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 passes through the points (3sqrt2, 2) and (6, 2sqrt(3)) , then find the value of e i.e., eccentricity of the given hyperbola.

If the foci of (x^2)/(a^2)-(y^2)/(b^2)=1 coincide with the foci of (x^2)/(25)+(y^2)/9=1 and the eccentricity of the hyperbola is 2, then a^2+b^2=16 there is no director circle to the hyperbola the center of the director circle is (0, 0). the length of latus rectum of the hyperbola is 12

Statement 1 : If from any point P(x_1, y_1) on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 , tangents are drawn to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1, then the corresponding chord of contact lies on an other branch of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 Statement 2 : From any point outside the hyperbola, two tangents can be drawn to the hyperbola.

If foci of (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 coincide with the foci of (x^(2))/(25)+(y^(2))/(16)=1 and eccentricity of the hyperbola is 3. then

The hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 passes through the point (2, ) and has the eccentricity 2. Then the transverse axis of the hyperbola has the length 1 (b) 3 (c) 2 (d) 4

The hyperbola (y^(2))/(a^(2))-(x^(2))/(b^(2)) =1 passes through the points (0, -2) and (sqrt(3). 4) . Find the value of e i.e., the eccentricity of the given hyperbola.

The hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 passes through the point (2,3 ) and has the eccentricity 2. Then the transverse axis of the hyperbola has the length (a)1 (b) 3 (c) 2 (d) 4

If a hyperbola passes through the foci of the ellipse (x^2)/(25)+(y^2)/(16)=1 . Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse and if the product of eccentricities of hyperbola and ellipse is 1 then the equation of hyperbola is (x^2)/9-(y^2)/(16)=1 b. the equation of hyperbola is (x^2)/9-(y^2)/(25)=1 c. focus of hyperbola is (5, 0) d. focus of hyperbola is (5sqrt(3),0)

If a hyperbola passes through the foci of the ellipse (x^2)/(25)+(y^2)/(16)=1 . Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse and if the product of eccentricities of hyperbola and ellipse is 1 then the equation of a. hyperbola is (x^2)/9-(y^2)/(16)=1 b. the equation of hyperbola is (x^2)/9-(y^2)/(25)=1 c. focus of hyperbola is (5, 0) d. focus of hyperbola is (5sqrt(3),0)