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

NTA Abhyas 2020 · Mathematics · Parabola

Let PQ be the focal chord of the parabola y ^ 2 = 4x. If the centre...

Let PQPQ be the focal chord of the parabola y2=4x.{y}^{2} = 4x. If the centre of the circle having PQPQ as its diameter lies on the line y=−45,y = \dfrac{- 4}{\sqrt{5}}, then the length (in units) of chord PQPQ is

Correct answer

7.2

Step-by-step solution

Let PP is (t2,2t)\left({t}^{2}, 2t\right) then QQ is \left(\dfrac{1}{{t}^{2}}, \dfrac{- 2}{t}\right)<br/>midpointof<br />mid point ofPQis(t2+1t22,2t−2t2)is \left(\dfrac{{t}^{2} + \dfrac{1}{{t}^{2}}}{2}, \dfrac{2t - \dfrac{2}{t}}{2}\right)
⇒t−1t=−45⇒t2+1t2−2=165⇒ t - \dfrac{1}{t} = \dfrac{- 4}{\sqrt{5}} ⇒ {t}^{2} + \dfrac{1}{{t}^{2}} - 2 = \dfrac{16}{5}
⇒t2+1t2+2=365⇒ {t}^{2} + \dfrac{1}{{t}^{2}} + 2 = \dfrac{36}{5}
⇒(t2+1)+(1t2+1)=7.2⇒ \left({t}^{2} + 1\right) + \left(\dfrac{1}{{t}^{2}} + 1\right) = 7.2
PS+SQ=7.2⇒PQ=7.2PS + SQ = 7.2 ⇒ PQ = 7.2
{ PS=a+at2,PQ=a+at2PS = a + {at}^{2}, PQ = a + \dfrac{a}{{t}^{2}} where a=1a = 1 }
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