Question

The transformed equation of \[ 4x^2-4xy+7y^2-24=0 \] in the new coordinate system when the axes are rotated through an angle \(\theta\) about the origin in the positive direction is \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. \] If \[ 0<\theta<\frac{\pi}{4}, \] then \[ 1+b^2\sin^2\theta= \]

• A. \(a^2\cos^2\theta\)
• B. \(a^2+\cos^2\theta\)
• C. \(a^2\sin^2\theta\)
• D. \(\dfrac{a^2}{\sin^2\theta}\)

Hint:

For second-degree equations, first eliminate the \(xy\)-term using \[ \boxed{\tan2\theta=\frac{2H}{A-B}}, \] then diagonalize the quadratic form to identify the semi-axes of the conic.

Answer 1

The Correct Option is C