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|Exercise Exercise 7.1|3 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.2|12 Videos

  • HIGHT AND DISTANCE

    CENGAGE|Exercise JEE Previous Year|3 Videos

  • INDEFINITE INTEGRATION

    CENGAGE|Exercise Question Bank|21 Videos

Similar Questions

Explore conceptually related problems

If x/a+y/b=sqrt(2) touches the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 , then find the eccentric angle theta of point of contact.

The sides A Ca n dA B of a A B C touch the conjugate hyperbola of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 . If the vertex A lies on the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 , then the side B C must touch (a)parabola (b) circle (c)hyperbola (d) ellipse

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.

A tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 cuts the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 at Pa n dQ . Show that the locus of the midpoint of P Q is ((x^2)/(a^2)+(y^2)/(b^2))^2=(x^2)/(a^2)-(y^2)/(b^2)dot

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 the chords of contact of tangents from two points (-4,2) and (2,1) to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 are at right angle, then find then find the eccentricity of the hyperbola.

From any point 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)=2. The area cut-off by the chord of contact on the asymptotes is equal to a/2 (b) a b (c) 2a b (d) 4a b

If any line perpendicular to the transverse axis cuts the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 and the conjugate hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 at points Pa n dQ , respectively, then prove that normal at Pa n dQ meet on the x-axis.

If the normal at one end of the latus rectum of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 passes through one end of the minor axis, then prove that eccentricity is constant.