ParabolaHardBloom L3

Question

Let there be two parabolas with the same axis, focus of each being exterior to the other and the latus recta 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 $a = b$
B.parabola if $a \neq b$
C.parabola $\forall a,b \in R$
D.none of these

Solution

Sol. [Key Concept: Midpoints of the vertical intercepts between two coaxial parabolas trace a straight line when $a = b$ and a parabola when $a \neq \ b$.]

$$\begin{matrix} & k = 2at \\ & 2h = at^{2} + \alpha - \frac{a^{2}t^{2}}{b} \\ & 2h = \left( a - \frac{a^{2}}{b} \right)\frac{k^{2}}{4a^{2}} + \alpha \\ & y^{2}\left( \frac{b - a}{4ab} \right)\frac{k^{2}}{4a^{2}} + \alpha = 2x \end{matrix}$$

If $a = b,\ 2x = \alpha,\ $ straight line

If

$a \neq b,y^{2}(b - a) + 4ab\alpha = 8abx$, Parabola

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