Let L be the point (t, 2) and M be a point on the y-axis such that the slope of LM is −t. Then, the locus of the midpoint of LM is a parabola whose latus rectum is
(a) 2
(b) 1/2
(c) 4
(d) 1/4
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Let L be the point (t, 2) and M be a point on the y-axis such that the slope of LM is −t. Then, the locus of the midpoint of LM is a parabola whose latus rectum is
(a) 2
(b) 1/2
(c) 4
(d) 1/4
Let L be the point (t, 2) and M be a point on the y-axis such that the slope of LM is −t. Then, the locus of the midpoint of LM is a parabola whose latus rectum is
(a) 2
(b) 1/2
(c) 4
(d) 1/4
Correct option (b) 1/2
Explanation :
Let M = (0, k) so that the slope of LM is
2 – k/t – 0 = -t
⇒ 2 – k = -t2 ….(1)
Let (x, y) be the midpoint of LM. Therefore
x = t/2 and y = 2 + k/2
Hence, from Eq. (1), we have
2y = 2 + k = 2 + (2 + t2)
= 4 + t2
= 4 + 4x2
y = 2 + 2x2
x2 = 1/2(y – 2)
Hence, the latus rectum is 1/2.