Question

An ellipse has lengths of major and minor axes as 6 and 2 respectively. If the center is at \( (5,6) \) and the major axis lies along the line \( x - y + 1 = 0 \), then the equation of the ellipse is:

• A. \( (x + y - 11)^2 + 9(x - y + 1)^2 = 18 \)
• B. \( (x + y + 11)^2 + 9(x - y - 1)^2 = 18 \)
• C. \( (x + y)^2 + 9(x - y)^2 = 18 \)
• D. \( (x - y - 11)^2 + 9(x + y + 1)^2 = 18 \)

Hint:

Ellipse with Inclined Axes}
Use rotation transformation: \( u = x + y, v = x - y \)
Replace variables in standard ellipse form
Remember: \( a = \frac{1}{2} \times \text{major axis length} \)

Answer 1

The Correct Option is A

Major axis length = 6 ⇒ semi-major axis \( a = 3 \) Minor axis length = 2 ⇒ semi-minor axis \( b = 1 \) Direction of major axis = along \( x - y + 1 = 0 \Rightarrow \) line rotated Transform coordinates: \[ u = x + y,\quad v = x - y \Rightarrow \text{Major axis along } v,\quad minor axis along \( u \) \] Standard ellipse: \[ \frac{v^2}{a^2} + \frac{u^2}{b^2} = 1 \Rightarrow \frac{(x - y + 1)^2}{9} + (x + y - 11)^2 = 1 \Rightarrow \text{Clear LCM: } (x + y - 11)^2 + 9(x - y + 1)^2 = 18 \]