Question

The length of the latus rectum of a conic \( 49y^2 - 16x^2 = 784 \) is

• A. \( \frac{49}{2} \)
• B. \( \frac{49}{\sqrt{2}} \)
• C. \( \frac{7}{\sqrt{2}} \)
• D. \( 7 \)

Hint:

The length of the latus rectum of a hyperbola is calculated using the formula \( L = \frac{2b^2}{a} \).

Answer 1

The Correct Option is A

Step 1: General equation of a conic.
The given equation is: \[ 49y^2 - 16x^2 = 784 \]
Divide the entire equation by 784 to simplify:
\[ \frac{49y^2}{784} - \frac{16x^2}{784} = 1 \] \[ \frac{y^2}{16} - \frac{x^2}{49} = 1 \] This is the standard form of a hyperbola.

Step 2: Identifying parameters for the hyperbola.
The standard form of a hyperbola is: \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] From the equation, we have \( a^2 = 16 \) and \( b^2 = 49 \). Therefore, \( a = 4 \) and \( b = 7 \).

Step 3: Formula for the length of the latus rectum.
The formula for the length of the latus rectum of a hyperbola is given by: \[ L = \frac{2b^2}{a} \]

Step 4: Substituting values.
Substituting the values \( b = 7 \) and \( a = 4 \): \[ L = \frac{2 \times 49}{4} = \frac{98}{4} = \frac{49}{2} \]

Step 5: Conclusion.
The length of the latus rectum is \( \frac{49}{2} \), so the correct answer is option (A).