JEE PYQ: Hyperbola Question 9
Question 9 - 2020 (05 Sep Shift 2)
If the line $y = mx + c$ is a common tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{64} = 1$ and the circle $x^2 + y^2 = 36$, then which one of the following is true?
(1) $c^2 = 369$
(2) $5m = 4$
(3) $4c^2 = 369$
(4) $8m + 5 = 0$
Show Answer
Answer: (3)
Solution
Tangent to hyperbola: $c^2 = 100m^2 - 64$. Tangent to circle: $c^2 = 36(1+m^2)$. Equating: $100m^2 - 64 = 36 + 36m^2$, so $64m^2 = 100$, $m^2 = \frac{100}{64}$. Then $c^2 = 36\left(1 + \frac{100}{64}\right) = \frac{36 \times 164}{64} = \frac{369}{4}$. So $4c^2 = 369$.
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