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If the line $3 x-m y+5=0$ is a tangent to the hyperbola $3 x^2-4 y^2=300$ then the square of the $Y-$ intercept made by this tangent line is
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Verified Answer
The correct answer is:
$\frac{15}{7}$
$3 x^2-4 y^2=300$
$$
\frac{x^2}{100}-\frac{y^2}{75}=1
$$
and $3 x-m y+5=0 \Rightarrow y=\frac{3}{m} x+\frac{5}{m}$ Now, for condition of tangents:
$$
\begin{aligned}
& \mathrm{c}^2=\mathrm{a}^2 \mathrm{~m}^2-\mathrm{n}^2 \Rightarrow \frac{25}{\mathrm{~m}^2}=100 \times \frac{9}{\mathrm{~m}^2}-75 \\
& \Rightarrow \mathrm{m}^2=\frac{35}{3}
\end{aligned}
$$
Square of $y$ - intercept of $x$-radius
$$
\mathrm{c}=\frac{25}{\mathrm{~m}^2}=25 \times \frac{3}{35}=\frac{15}{7}
$$
$$
\frac{x^2}{100}-\frac{y^2}{75}=1
$$
and $3 x-m y+5=0 \Rightarrow y=\frac{3}{m} x+\frac{5}{m}$ Now, for condition of tangents:
$$
\begin{aligned}
& \mathrm{c}^2=\mathrm{a}^2 \mathrm{~m}^2-\mathrm{n}^2 \Rightarrow \frac{25}{\mathrm{~m}^2}=100 \times \frac{9}{\mathrm{~m}^2}-75 \\
& \Rightarrow \mathrm{m}^2=\frac{35}{3}
\end{aligned}
$$
Square of $y$ - intercept of $x$-radius
$$
\mathrm{c}=\frac{25}{\mathrm{~m}^2}=25 \times \frac{3}{35}=\frac{15}{7}
$$
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