Question

The line \( y = mx + C \) will be tangent to the ellipse \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) if \( C \) is equal to

• A. \( \frac{3}{m} \)
• B. \( \sqrt{9m^2 + 4} \)
• C. \( \sqrt{1 + m^2} \)
• D. \( \sqrt{4m^2 + 9} \)

Hint:

In problems involving tangents to conic sections, use the distance formula from the center to the line to find the condition for tangency.

Answer 1

The Correct Option is B

The equation of the tangent to the ellipse can be written as \( y = mx + C \). The distance from the center of the ellipse (which is at the origin) to this line is equal to the semi-major axis, which is 3. Using the formula for the distance from a point to a line: \[ {Distance} = \frac{|C|}{\sqrt{1 + m^2}} = 3 \] Thus: \[ |C| = 3 \sqrt{1 + m^2} \] Since we are given that the line is tangent, the correct formula for \( C \) is: \[ C = \sqrt{9m^2 + 4} \]