Question

The line \( 21x + 5y = k \) touches the hyperbola \( 7x^2 - 5y^2 = 232 \), then \( k \) is:

• A. \( 116 \)
• B. \( 232 \)
• C. \( 58 \)
• D. \( 110 \)

Hint:

For a line to be tangent to a hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the equation of the tangent should satisfy the standard tangency condition.

Answer 1

The Correct Option is A

Step 1: Standard form of the given hyperbola 
The given hyperbola equation is: \[ 7x^2 - 5y^2 = 232. \] Dividing by 232 to convert into standard form: \[ \frac{x^2}{\frac{232}{7}} - \frac{y^2}{\frac{232}{5}} = 1. \] Thus, the standard form is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \] where \( a^2 = \frac{232}{7} \) and \( b^2 = \frac{232}{5} \).

Step 2: Condition for tangency 
The equation of a general tangent to the hyperbola is: \[ \frac{x}{a^2} A - \frac{y}{b^2} B = 1. \] Comparing with the given line \( 21x + 5y = k \), we use the condition of tangency: \[ \sqrt{\frac{A^2}{a^2} - \frac{B^2}{b^2}} = 1. \] Substituting values: \[ \sqrt{\frac{21^2}{\frac{232}{7}} - \frac{5^2}{\frac{232}{5}}} = 1. \] 

Step 3: Solving for \( k \) 
Solving the equation gives: \[ k = 116. \] 

Step 4: Conclusion 
Thus, the final answer is: \[ \boxed{116}. \]