Question

If the equation of an ellipse is \(3x^2+2y^2+6x-8y+5=0\), then which of the following are true?

• A. \(e=\frac{1}{\sqrt{3}}\)
• B. centre is \((-1,2)\)
• C. foci are \((-1,1)\) and \((-1,3)\)
• D. All of the above

Hint:

Convert ellipse to standard form by completing squares. Then \(c^2=a^2-b^2\) and foci lie along major axis from centre.

Answer 1

The Correct Option is C

Step 1: Rewrite ellipse equation by completing squares.
Given:
\[ 3x^2+2y^2+6x-8y+5=0 \] Group terms:
\[ 3(x^2+2x)+2(y^2-4y)+5=0 \] Complete square:
\[ x^2+2x=(x+1)^2-1 \] \[ y^2-4y=(y-2)^2-4 \] Substitute:
\[ 3[(x+1)^2-1]+2[(y-2)^2-4]+5=0 \] \[ 3(x+1)^2-3+2(y-2)^2-8+5=0 \] \[ 3(x+1)^2+2(y-2)^2-6=0 \] \[ 3(x+1)^2+2(y-2)^2=6 \] Divide by 6:
\[ \frac{(x+1)^2}{2}+\frac{(y-2)^2}{3}=1 \] Step 2: Identify axes.
Here:
\[ a^2=3,\quad b^2=2 \] Major axis along \(y\)-direction.
Step 3: Find foci.
\[ c^2=a^2-b^2=3-2=1 \Rightarrow c=1 \] Centre is \((-1,2)\).
Since major axis along \(y\), foci are:
\[ (-1,2\pm 1)=(-1,1),(-1,3) \] Step 4: Match correct statement.
Statement (C) is correct.
Final Answer: \[ \boxed{\text{foci are }(-1,1)\text{ and }(-1,3)} \]