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Exercise 7.3 · Q73

Q.Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola given by x=2sec⁡θ, y=23tan⁡θx = 2\sec\theta,\ y = 2\sqrt3\tan\theta.

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Comparing x=2sec⁡θ, y=23tan⁡θx=2\sec\theta,\,y=2\sqrt3\tan\theta with the standard parametric form x=asec⁡θ, y=btan⁡θx=a\sec\theta,\,y=b\tan\theta of x2a2−y2b2=1\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1: a=2, b=23a=2,\,b=2\sqrt3, so a2=4, b2=12a^2=4,\,b^2=12.

e2=1+124=4⇒e=2e^2=1+\dfrac{12}{4}=4 \Rightarrow e=2.

  • Transverse axis =4=4; conjugate axis =43=4\sqrt3.
  • Foci (±ae,0)=(±4,0)(\pm ae,0)=(\pm4,0).
  • Directrices x=±ae=±1x=\pm\dfrac{a}{e}=\pm1. …
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