Let A be the point (t, 2) and B be the point on the y-axis such that the slope of AB is -t. Then, the locus of the midpoint of AB, as t varies over all real numbers, is
(a) y = 2 - 2x2
(b) x2 + y - 1 = 0
(c) y = 1 + x2
(d)2x2 - y + 2 = 0
Correct option (d)2x2 - y + 2 = 0
Explanation :
See Fig. Equation of the line AB is
Let M(x, y) be the midpoint of AB. Therefore
x= t + 0/2, y = 2+(2 + t2)/2
t = 2x, 2y = 4 + t2 = 4 + 4x2
Thus, the locus of M is y = x2 + 2x + 2x2.
