If l denotes the semi-latus rectum of the parabola y2=4ax and SP and SQ denote the segments of any focal chord PQ, S being the focus, then SP, l and SQ are in the relation
A
A.P
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B
G.P
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C
H.P
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D
l2=SP2+SQ2
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Solution
The correct option is C H.P Let PQ be focal chord of the parabola y2=4ax with focus S(a,0) P(at21,2at1),Q(at22,2at2) are such that SP and SQ are segments of focal chord ⇒t1t2=−1 SP=a+at21=a(1+t21) SQ=a(t22+1)=a(1t21+1)=a(1+t21t21)[∵t1t2=−1] ∴(1SP+1SQ)=1a(t21+1)+t21a(t21+1)=1a[t21+1t21+1]=1a ⇒SQ+SPSQ.SQ=1a⇒2SP.SQSQ+SP=2a=l (semi-latus rectum=2a) ∴ Hence SP,l,SQ are in HP