Question

Let the foci of a hyperbola be \( (1, 14) \) and \( (1, -12) \). If it passes through the point \( (1, 6) \), then the length of its latus-rectum is:

• A. \( \frac{25}{6} \)
• B. \( \frac{24}{5} \)
• C. \( \frac{288}{5} \)
• D. \( \frac{144}{5} \)

Hint:

The length of the latus-rectum of a hyperbola is determined by the formula \( \frac{2a^2}{b} \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

Answer 1

The Correct Option is C

The hyperbola's foci are located at \( (1, 14) \) and \( (1, -12) \). The distance between the foci is \( 2c \), which is given as \( 2c = \sqrt{(1-1)^2 + (14-(-12))^2} = \sqrt{0^2 + 26^2} = 26 \). Therefore, \( c = 13 \). We are also given that \( be = 13 \) and \( b = 5 \). Using the formula \( a^2 = b^2 (e^2 - 1) \), we rearrange it to \( a^2 = b^2 e^2 - b^2 \). Since \( be = 13 \), \( b^2 e^2 = (be)^2 = 13^2 = 169 \). Substituting the values, we get \( a^2 = 169 - 5^2 = 169 - 25 = 144 \). The length of the latus-rectum is calculated using the formula \( \ell (LR) = \frac{2a^2}{b} \), which results in \( \ell (LR) = \frac{2 \times 144}{5} = \frac{288}{5} \).