JEE MAIN - Mathematics (2024 - 6th April Morning Shift - No. 22)
Let a conic $$C$$ pass through the point $$(4,-2)$$ and $$P(x, y), x \geq 3$$, be any point on $$C$$. Let the slope of the line touching the conic $$C$$ only at a single point $$P$$ be half the slope of the line joining the points $$P$$ and $$(3,-5)$$. If the focal distance of the point $$(7,1)$$ on $$C$$ is $$d$$, then $$12 d$$ equals ________.
Answer
75
Explanation
As per given condition
$$\begin{gathered} \frac{d y}{d x}=\frac{y+5}{2(x-3)} \\ \Rightarrow \ln (y+5)=\frac{1}{2} \ln (x-3)+c \\ \text { Passes through }(4,-2) \Rightarrow \ln 3=\frac{1}{2} \ln 1+c \\ \Rightarrow c=\ln 3 \end{gathered}$$
$$\Rightarrow$$ Curve is $$(y+5)^2=9(x-3)$$
Focal distance of $$(7,1)=\frac{9}{4}+4=\frac{25}{4}=d$$
$$12 d=75$$
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