KCET · Maths · Hyperbola
If \(\frac{x^{2}}{36}-\frac{y^{2}}{k^{2}}=1\) is a hyperbola, then which of the following statements can be true?
- A \((-3,1)\) lies on the hyperbola
- B \((3,1)\) lies on the hyperbola
- C \((10,4)\) lies on the hyperbola
- D \((5,2)\) lies on the hyperbola
Answer & Solution
Correct Answer
(C) \((10,4)\) lies on the hyperbola
Step-by-step Solution
Detailed explanation
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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