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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Hyperbola

NTA Abhyas 2020 · 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)<br/>Hence,6 < 5 - \left(-5\right)<br />Hence, the locus is a hyperbola by definition.
Let's assume the eccentricity, semi-transverse axis and the semi-conjugate axis of the hyperbola ise, a, and brespectively,then</p><p>respectively, then</p><p>2ae = 10{Distance between focii \left(-5,0\right) & \left(5,0\right) = 10 = 2ae }
& 2a=6⇒e=106=532a = 6 ⇒ e = \dfrac{10}{6} = \dfrac{5}{3}
⇒1+b2a2=⇒ 1 + \dfrac{b^{2}}{a^{2}} = e^{2} = 259⇒b2a2=259−\dfrac{25}{9}⇒ \dfrac{b^{2}}{a^{2}} = \dfrac{25}{9} -1 = 169\dfrac{16}{9}
⇒2b2a=2×169⇒ \dfrac{2b^{2}}{a} =2 \times \dfrac{16}{9} \timesa = 323=length\dfrac{32}{3}=length of the latus rectum

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