Let there be two parabolas with the same axis, focus of each being exterior to the other and the latus rectam being 4a and 4b. The locus of the middle points of the intercepts between the parabolas made on the lines parallel to the common axis is a:

Options

A. `"straight line if a"gtb`
B. `"parabola if a"inb`
C. `"parabola for all a, b"`
D. `"ellipse, if "b gt a`

Answer

B

Explanation

Let the two parabolas be `y^(2)=4a(x-1)" and "y^(2)=-4b(x+m)`. Further, let y = h be a line parallel to the common axis. This intersects the two parabolas at
`A(h^(2)/(4a)+1, )" and B "(-h^(2)/(4b)-m, h)" respectively".`
Let `P(alpha, beta)` be the mid-point of AB. Then,
`2alpha=h^(2)/4(1/a-1/b)+l-m" and " beta=h`
`rArr" "2alpha=beta^(2)/4(1/a-1/b)+i-m`

Thus, the locus of `(alpha, beta)` is
`2x=y^(2)/4(1/a-1/b)+l-m`
Clearly, it represents a parabola if `a in b`.
If a = b, then we obtain
2x=l-m which represetnts a lien.