Sequences & Series
Parabolas intersecting a line — GP/AP of coefficients
nta_pyq_2023_jan
Grade 11
Question:
The parabolas $ax^2 + 2bx + cy = 0$ and $dx^2 + 2ex + fy = 0$ intersect on the line $y = 1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then
d, e, f are in A.P.
\dfrac{d}{a}, \dfrac{e}{b}, \dfrac{f}{c} are in G.P.
\dfrac{d}{a}, \dfrac{e}{b}, \dfrac{f}{c} are in A.P.
d, e, f are in G.P.
Step-by-Step Solution
Key Concept: Since $a, b, c$ in GP means $b^2 = ac$. Parabola 1: $ax^2 + 2bx + c = 0$ (at $y=1$) has double root $x = -\sqrt{c/a}$. Substitute this common root into the second parabola equation.
Common point: $x = -\sqrt{c/a}$. Substituting: $\frac{dc}{a} - 2e\sqrt{\frac{c}{a}} + f = 0 \Rightarrow \frac{d}{a} + \frac{f}{c} = \frac{2e}{b}$. So $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.
Correct Answer: 3