Question

What conic does \(13x^2 - 18xy + 37y^2 + 2x + 14y - 2 = 0\) represent?

• A. Circle
• B. Ellipse
• C. Parabola
• D. Hyperbola

Answer 1

The Correct Option is B

Step 1: Write the general second-degree equation of a conic. \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] From the given equation: \[ 13x^2 - 18xy + 37y^2 + 2x + 14y - 2 = 0 \] we identify: \[ A = 13,\quad B = -18,\quad C = 37 \] Step 2: Use the discriminant to identify the conic. The discriminant is: \[ \Delta = B^2 - 4AC \] \[ \Delta = (-18)^2 - 4(13)(37) \] \[ \Delta = 324 - 1924 \] \[ \Delta = -1600 \] Step 3: Interpret the result.

If \(B^2 - 4AC = 0\) \(\Rightarrow\) Parabola
If \(B^2 - 4AC>0\) \(\Rightarrow\) Hyperbola
If \(B^2 - 4AC<0\) \(\Rightarrow\) Ellipse (or circle if \(A=C\) and \(B=0\))
Here, \[ B^2 - 4AC = -1600<0 \] and \(A \neq C\), so the conic is an ellipse. Step 4: Final conclusion. The given equation represents an \(\boxed{\text{Ellipse}}\).