Conic-Sections Question 555
Question: Through the vertex O at a parabola $ y^{2}=4x, $ chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ is
Options:
A) $ y^{2}=2x+8 $
B) $ y^{2}=x+8 $
C) $ y^{2}=2x-8 $
D) $ y^{2}=x-8 $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Given parabola is $ y^{2}=4x $ ? (1) Let $ P\equiv ( t^2_1,2t_1 ) $ and $ Q\equiv ( t^2_2,2t_2 ) $ Slope of $ OP=\frac{2t_1}{t^2_1}=\frac{2}{t_1} $ and slope of $ OQ=\frac{2}{t_2} $ Since $ OP\bot OQ, $
$ \therefore \frac{4}{t_1t_2}=-1 $ or $ t_1t_2=-4 $ ? (2) Let $ R(h,k) $ be the middle point of PQ, then $ h=\frac{t_1^{2}+t_2^{2}}{2} $ ? (3) and $ k=t_1+t_2 $ ? (4) From (4), $ k^{2}=t^2_1+t^2_2+2t_1t_2=2h-8 $ [From (2) and (3)] Hence locus of $ R(h,k) $ is $ y^{2}-2x-8. $
BETA
