Question

The difference between the focal distances of any point on the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text{ is } 6. \text{ If } (\sqrt{13}, k) \text{ is an end point of a latus rectum of this hyperbola, then } k = \]

• A. \( \frac{9}{2} \)
• B. \( \frac{8}{3} \)
• C. \( \frac{9}{3} \)
• D. \( \frac{4}{3} \)

Hint:

Remember that for hyperbolas, the length of the latus rectum is \( \frac{2b^2}{a} \) and the relationship between the focal distance and the vertices is \( c^2 = a^2 + b^2 \).

Answer 1

The Correct Option is D

For a hyperbola, the difference between the focal distances of any point on the hyperbola is given by \( 2a \). So, we are given that the difference is 6, so \( 2a = 6 \), which gives \( a = 3 \).
Now, we are given that \( (\sqrt{13}, k) \) is an end point of the latus rectum of the hyperbola. The equation of the hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), and the length of the latus rectum is given by \( \frac{2b^2}{a} \).
We know that \( c^2 = a^2 + b^2 \) for hyperbolas. Using the given information and solving for \( k \), we find that \( k = \frac{4}{3} \).
Therefore, the correct answer is \( \frac{4}{3} \).

Answer 2

The given hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). For a hyperbola, the focal distance of any point \(P(x_1, y_1)\) is \(|e \cdot x_1|\), where \(e\) is the eccentricity. The eccentricity \(e\) is given by:

\[e = \frac{\sqrt{a^2 + b^2}}{a}\]

The difference between the focal distances of any point on the hyperbola is \(2a\), and it's given as 6:

\[2a = 6 \implies a = 3\]

The coordinates of the end points of the latus rectum are \(\left(\pm ae, \pm \frac{b^2}{a}\right)\). Given \((\sqrt{13}, k)\) is one of them:

\[\text{Since } x = \sqrt{13}, \sqrt{13} = ae\]

\[e = \frac{\sqrt{13}}{3}\]

Substituting \(e\) in \(b^2 = a^2(e^2 - 1)\):

\[b^2 = 9 \left(\left(\frac{\sqrt{13}}{3}\right)^2 - 1\right) = 9 \left(\frac{13}{9} - 1\right) = 9 \times \frac{4}{9} = 4\]

The \(y\)-coordinate of the end point of the latus rectum, \(k\), is:

\[k = \pm \frac{b^2}{a} = \pm \frac{4}{3}\]

Thus, the value of \(k\) is \(\frac{4}{3}\).