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Joint Entrance Examination
Mathematics
Parabola
Question 1
Question 1 of 198
Medium
MCQ
Let O be the vertex of the parabola
y
2
=
4
x
y^2=4 x
y
2
=
4
x
and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C , then the length of its latus rectum is :
A
1
B
2
C
4
D
8
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Official solution
t
1
t
2
=
−
4
2
h
=
t
2
1
+
t
2
2
2
k
=
2
(
t
1
+
t
2
)
2
h
=
k
2
−
2
t
1
t
2
2
h
=
k
2
+
8
k
2
=
2
h
−
8
=
2
(
(
h
−
4
)
=
2
\begin{gathered} {t}_{1}{t}_{2}=-4 \\ 2h={t}_{2}^{1}+{t}_{2}^{2}\, 2k=2({t}_{1}+{t}_{2}) \\ 2h={k}^{2}-2{t}_{1}{t}_{2} \\ 2h={k}^{2}+8 \\ {k}^{2}=2h-8 \\ =2((h-4) \\ =2 \end{gathered}
t
1
t
2
=
−
4
2
h
=
t
2
1
+
t
2
2
2
k
=
2
(
t
1
+
t
2
)
2
h
=
k
2
−
2
t
1
t
2
2
h
=
k
2
+
8
k
2
=
2
h
−
8
=
2
((
h
−
4
)
=
2
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