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JEE Main
Exam
JEE Main
Subject
Mathematics
Chapter
Hyperbola

JEE Main · Mathematics · Hyperbola

The locus of a point which moves such that the difference of its...

The locus of a point which moves such that the difference of its distances from the points (5,0)\left(5,0\right) and (−5,0)\left(- 5,0\right) is 66 units is a conic, whose length of the latus rectum (in units) is equal to

Options

  1. A44
  2. B163\dfrac{16}{3}
  3. C88
  4. D323\dfrac{32}{3}

Correct answer

D. 323\dfrac{32}{3}

Step-by-step solution

Because 6<5−(−5)6 < 5 - \left(-5\right)
Hence, the locus is a hyperbola by definition.
Let's assume the eccentricity, semi-transverse axis and the semi-conjugate axis of the hyperbola is e,a,e, a, and bb respectively, then

2ae=102ae = 10 {Distance between focii \left(-5,0\right) & \left(5,0\right) = 10 = 2ae}<br/>&2a = 6 ⇒ e = 106=53<br/>\dfrac{10}{6} = \dfrac{5}{3}<br/>⇒ 1 + b2a2=\dfrac{b^{2}}{a^{2}} =e^{2} = 259\dfrac{25}{9}⇒ b2a2=259−\dfrac{b^{2}}{a^{2}} = \dfrac{25}{9} -1 = 169<br/>\dfrac{16}{9}<br/>⇒ 2b2a=\dfrac{2b^{2}}{a} =2 \times \dfrac{16}{9} \timesa = \dfrac{32}{3}=lengthofthelatusrectum</p>length of the latus rectum</p>

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