Question

If \( S = \frac{x^2}{k - 7} - \frac{y^2}{11 - k} = 0, k \in \mathbb{R}, k \neq 7,11 \), then which one of the following statements is incorrect?

• A. \( S = 0 \) represents a circle with radius \( \sqrt{2} \), when \( k = 9 \)
• B. \( S = 0 \) represents an ellipse with eccentricity \( \frac{\sqrt{2}}{3} \), when \( k = 10 \)
• C. \( S = 0 \) represents a hyperbola with eccentricity \( \frac{\sqrt{6}}{5} \), when \( k = 12 \)
• D. \( S = 0 \) represents a hyperbola with eccentricity \( \frac{\sqrt{3}}{2} \), when \( k = 13 \)

Hint:

For conic sections, identify the type based on the sign and values of the denominators. The eccentricity can be calculated based on the coefficients.

Answer 1

The Correct Option is B

We are given the equation of a conic: \[ \frac{x^2}{k - 7} - \frac{y^2}{11 - k} = 0 \] This represents a conic whose type depends on the value of \( k \). The nature of the conic (circle, ellipse, hyperbola) depends on the signs of the terms in the equation. Step 1: Conditions for a circle, ellipse, or hyperbola - Circle: \( \frac{x^2}{r^2} + \frac{y^2}{r^2} = 0 \) — when \( k = 9 \) - Ellipse: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \) — when \( k = 10 \) - Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \) — when \( k = 12 \) Step 2: Check the values for each case - For \( k = 9 \), the equation represents a circle with radius \( \sqrt{2} \), which is correct. - For \( k = 10 \), the equation represents an ellipse, but the eccentricity does not match the given value \( \frac{\sqrt{2}}{3} \), so this statement is incorrect. - For \( k = 12 \), the equation represents a hyperbola with eccentricity \( \frac{\sqrt{6}}{5} \), which is correct. - For \( k = 13 \), the equation represents a hyperbola with eccentricity \( \frac{\sqrt{3}}{2} \), which is also correct. Thus, the correct answer is option (2), which is incorrect.