Let 'L' be the point `(t, 2) and 'M'` be a point on the y axis such that `'L M'` has slope `-'t'`. Then the locus of the mid point of `'L M', as 't'` varies over all real values, is a parabola, whose

Options

A. vertex is (0, 2)
B. lengths of latus-rectum is 2
C. focus is `(0,(17)/(8))`
D. equation of directrix is 8y-15=0

Answer

A, C, D

Explanation

Now, slope of `LM=(2-alpha)/(t-0)=-t` (given)
`rArr alpha-2=t^(2)rArr alpha = 2 + t^(2)`
Let mid point of LM is (h, k)
Now, `2h = t and 2k=alpha+2=t^(2)+4`
`:.` On eliminating t, we get
`k=2(h^(2)+1)`
`:.` Locus of (h, k) is `y=2(x^(2)+1)rArr x^(2)=(1)/(2)(y-2)`
Now, verify alternative.