JEE PYQ: Parabola Question 11
Question 11 - 2020 (07 Jan Shift 1)
If $y = mx + 4$ is a tangent to both the parabolas, $y^2 = 4x$ and $x^2 = 2by$, then $b$ is equal to:
(1) $-32$
(2) $-64$
(3) $-128$
(4) $128$
Show Answer
Answer: (3)
Solution
Tangent to $y^2 = 4x$: $y = mx + \frac{1}{m}$. Since $y = mx + 4$: $\frac{1}{m} = 4 \Rightarrow m = \frac{1}{4}$. Line $y = \frac{1}{4}x + 4$ is also tangent to $x^2 = 2by$. Substituting: $x^2 = 2b\left(\frac{x+16}{4}\right) \Rightarrow 2x^2 - bx - 16b = 0$. $D = 0$: $b^2 + 32 \times 4b = 0 \Rightarrow b = -128$.
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