Parabola
Parabola
nta_abhyas_2025
Grade 11
Question:
Focus of parabola $y^2 = 4ax$ is $(a, 0)$. Focus of parabola in option (A) is $(0b, 0)$. Focus of parabola in option (B) is $(2b, 0)$. Focus of parabola in option (C) is $(30b, 0)$. Focus of parabola in option (D) is $(k, 0)$.
(0b, 0)
(2b, 0)
(30b, 0)
(k, 0)
Step-by-Step Solution
Key Concept: The chord of contact and properties of normals at symmetric points on a parabola help determine the parameter relationship.
Let $A = (p, q)$ and $B = (p, -q)$. From $q^2 = 4a(p - k)$ we get $q^2 = 4a(p - k)$ ... (1). Now for chord $C_1$: $\frac{dy}{dx}|_{(p,q)} = \frac{2a}{q}$ and $\frac{dy}{dx}|_{(p,-q)} = \frac{2a}{-q}$, giving slopes $\frac{2a}{q}$ & $-\frac{2a}{q}$. Putting values from (i) & (iii) in the normal equation: $4q^2 = 4a(\frac{a+k}{2} - a) = a = \frac{k-a}{2} ⟹ k = 3a$.
Correct Answer: 4