Question

The value of \( \lambda \) does the line \( y = x + \lambda \) touches the ellipse \( 9x^2 + 16y^2 = 144 \) is:

• A. \( \pm 2\sqrt{2} \)
• B. \( \pm \sqrt{5} \)
• C. 5
• D. \( \pm 5 \)

Hint:

To find the value of \( \lambda \) where a line is tangent to an ellipse, substitute the line’s equation into the ellipse equation and solve for the point of tangency.

Answer 1

The Correct Option is A

Step 1: The equation of the ellipse is \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \). The equation of the line is \( y = x + \lambda \).
Step 2: Substitute \( y = x + \lambda \) into the equation of the ellipse. After simplifying and solving for \( \lambda \), we find \( \lambda = \pm 2\sqrt{2} \).

Final Answer: \[ \boxed{\pm 2\sqrt{2}} \]