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If $\frac{x^{2}}{36}-\frac{y^{2}}{k^{2}}=1$ is a hyperbola, then which of the following statements can be true?
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2417 Upvotes
Verified Answer
The correct answer is:
$(10,4)$ lies on the hyperbola
Given, $\frac{x^{2}}{36}-\frac{y^{2}}{k^{2}}=1$
$$
\begin{array}{ll}
\Rightarrow & \frac{y^{2}}{k^{2}}=\frac{x^{2}}{36}-1 \\
\Rightarrow & k^{2}=\frac{36 y^{2}}{x^{2}-36}
\end{array}
$$
$$
k^{2}>0
$$
If $\quad x^{2}-36>0$
$\Rightarrow \quad x^{2}>36$
This is true only for point $(10,4)$. So, $(10,4)$ lies on the hyperbola.
$$
\begin{array}{ll}
\Rightarrow & \frac{y^{2}}{k^{2}}=\frac{x^{2}}{36}-1 \\
\Rightarrow & k^{2}=\frac{36 y^{2}}{x^{2}-36}
\end{array}
$$
$$
k^{2}>0
$$
If $\quad x^{2}-36>0$
$\Rightarrow \quad x^{2}>36$
This is true only for point $(10,4)$. So, $(10,4)$ lies on the hyperbola.
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