Question

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

For a rotated ellipse, first identify the directions of the major and minor axes. Then express the equation in coordinates measured along those directions.

Answer 1

The Correct Option is A

Step 1: Determine the semi-axes.
Length of major axis \[ =6 \] and length of minor axis \[ =2. \] Therefore, \[ a=3,\qquad b=1. \]

Step 2: Choose coordinates along the principal axes.
The major axis is along \[ x-y+1=0, \] whose direction is parallel to \[ x-y=0. \] Hence take \[ U=x-y+1. \] The perpendicular direction is \[ V=x+y-11, \] since the centre is \[ (5,6). \]

Step 3: Write the ellipse equation.
The ellipse in rotated coordinates is \[ \frac{V^2}{a^2}+\frac{U^2}{b^2}=1. \] Thus, \[ \frac{(x+y-11)^2}{9} + (x-y+1)^2 =1. \] Multiplying by \(18\), \[ (x+y-11)^2+9(x-y+1)^2=18. \]

Step 4: Final conclusion.
Therefore, the required ellipse is \[ \boxed{(x+y-11)^2+9(x-y+1)^2=18}. \]