Let A be a symmetric 2 ( 2 matrix and let k be a scalar. Prove that

Question:

Let A be a symmetric 2 ( 2 matrix and let k be a scalar. Prove that the graph of the quadratic equation xT Ax = k is
(a) A hyperbola if k ≠ 0 and det A < 0
(b) An ellipse, circle, or imaginary conic if k ≠ 0 and det A > 0
(c) A pair of straight lines or an imaginary conic if k ≠ 0 and det A = 0
(d) A pair of straight lines or a single point if k = 0 and det A ≠ 0
(e) A straight line if k = 0 and det A = 0 [Hint: Use the Principal Axes Theorem.]