Home
Class 12
MATHS
If P Q is a double ordinate of the hyper...

If `P Q` is a double ordinate of the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` such that `O P Q` is an equilateral triangle, `O` being the center of the hyperbola, then find the range of the eccentricity `e` of the hyperbola.

Text Solution

Verified by Experts


Let the hyperbola be `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`
and any double ordinate PQ be such that `P-=(a sec theta,b tan theta)`.
`therefore" "Q-=(a sec theta, -b tan theta)`
According to the question, triangle OPQ is equilateral.
`therefore" "tan30^(@)=(b tan theta)/(a sec theta)`
`rArr" "3(b^(2))/(a^(2))="cosec"^(2)theta`
`rArr" "3(e^(2)-1)="cosec"^(2)theta`
Now, `"cosec"^(2)thetage1`
`rArr" "3(e^(2)-1)ge1`
`rArr" "e^(2)ge(4)/(3)`
`rArr" "ege(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|25 Videos

Similar Questions

Explore conceptually related problems

If PQ is a double ordinate of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 such that OPQ is an equilateral triangle,O being the center of the hyperbola, then find the range of the eccentricity e of the hyperbola.

If PQ is a double ordinate of the hyperbola such that OPQ is an equilateral triangle, being the centre of the hyperbola, then eccentricity e of the hyperbola satisfies

If the eccentricity of a hyperbola is 2, then find the eccentricity of its conjugate hyperbola.

If the latus rectum of a hyperbola forms an equilateral triangle with the vertex at the center of the hyperbola,then find the eccentricity of the hyperbola.

AB is double ordinate of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 such that Delta AOB( where 'O' is the origin) is an equilateral triangle,then the eccentricity e of hyperbola satisfies:

if the eccentricity of the hyperbola is sqrt2 . then find the general equation of hyperbola.

The eccentricity of the hyperbola x ^(2) - y^(2) =25 is

Find the eccentricity of the hyperbola 9y^(2)-4x^(2)=36

If the slope of a tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 is 2sqrt(2) then the eccentricity e of the hyperbola lies in the interval