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Solve the following.
If y - 2x - k = 0 touches the conic 3x² - 5y² = 15, find the value of k.
If $y - 2x - k = 0$ touches the conic $3x^2 - 5y^2 = 15$, find the value of $k$.
Answer
Answer
Write the conic as $\frac{x^2}{5} - \frac{y^2}{3} = 1$, so $a^2 = 5$, $b^2 = 3$.
The line $y = 2x + k$ has $m = 2$ and touches the hyperbola if $k^2 = a^2m^2 - b^2$.
$k^2 = 5(4) - 3 = 17$
Hence $k = \pm\sqrt{17}$.
Final answer: $\pm\sqrt{17}$