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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Hyperbola

NDA 2023 · Mathematics · Hyperbola

Consider the following statements in respect of hyperbola (x^2)/(...

Consider the following statements in respect of hyperbola x2cos⁡2θ−y2sin⁡2θ=11.Thetwofociareindependentof0.2.Theeccentricityis\frac{x^2}{\cos ^2 \theta}-\frac{y^2}{\sin ^2 \theta}=1 1. The two foci are independent of 0 . 2. The eccentricity is\sec \theta.3.Thedistancebetweenthetwofociis2units.Howmanyofthestatementsgivenabovearecorrect?. 3. The distance between the two foci is 2 units. How many of the statements given above are correct?

Options

  1. AOnly one
  2. BOnly two
  3. CAll three
  4. DNone

Correct answer

C. All three

Step-by-step solution

Given the equation x2cos⁡2θ−y2sin⁡2θ=1\frac{x^2}{\cos ^2 \theta}-\frac{y^2}{\sin ^2 \theta}=1 Since C2=cos⁡2θ+sin⁡2θ=1C^2=\cos ^2 \theta+\sin ^2 \theta=1 So focus =(0±1,0)=(±1,0)e=ca=1cos⁡θ=sec⁡θ=(0 \pm 1,0)=( \pm 1,0)e=\frac{c}{a}=\frac{1}{\cos \theta}=\sec \theta Distance between the two foci =1+1=2=1+1=2
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