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

NTA Abhyas 2020 · Mathematics · Parabola

Chord joining two distinct points P (a, 4b) and Q (c, - (16)/(b))...

Chord joining two distinct points PP\left(a, 4b\right)andandQ \left(c, - \dfrac{16}{b}\right) (both are variable points) on the parabola y2=16xy^{2} = 16x always passes through a fixed point \left(\alpha, \beta\right).Then, which of the following statements is correct?

Options

  1. Aα+β=2\alpha + \beta = 2
  2. Bα−β=4\alpha - \beta = 4
  3. C∣α∣+∣β∣=8\left|\alpha\right| + \left|\beta\right| = 8
  4. D∣α∣=∣β∣\left|\alpha\right| = \left|\beta\right|

Correct answer

B. α−β=4\alpha - \beta = 4

Step-by-step solution

Let, P=(a,4b)=(4t12,8t1)Q=(c,−16b)=(4t22,8t2)P = \left(a, 4b\right) = \left(4{t}_{1}^{2}, 8{t}_{1}\right) Q = \left(c, - \dfrac{16}{b}\right) =\left(4{t}_{2}^{2}, 8{t}_{2}\right)
⇒t1=b2and⇒ {t}_{1} = \dfrac{b}{2} and{t}_{2} = - \dfrac{2}{b}<br/><br />⇒ {t}_{1}{t}_{2} = - 1<br/><br />⇒ChordChordPQisafocalchordthatpassesthroughthefocus<br/>is a focal chord that passes through the focus<br />⇒ \left(\alpha, \beta\right) = \left(4,0\right)
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