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Multivariable Calculus · Question of the Day

Multivariable Calculus Question of the Day

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Saturday, October 10, 2026

The level curves of the surface z=f(x,y)z = f(x, y) are described by the equation 9x2+4y2−18x+16y=k9x^2 + 4y^2 - 18x + 16y = k for some constant kk. Which of the following statements accurately describes this surface?

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The level curves of the surface z=f(x,y)z = f(x, y) are described by the equation 9x2+4y2−18x+16y=k9x^2 + 4y^2 - 18x + 16y = k for some constant kk. Which of the following statements accurately describes this surface?

  1. The surface is a hyperbolic paraboloid with a saddle point at (1,−2)(1, -2).
  2. The surface is an elliptic paraboloid with a minimum at (1,−2)(1, -2). (correct answer)
  3. The surface is a circular cone with vertex at (1,−2)(1, -2).
  4. The surface is a tilted plane with parallel level curves.

Explanation: To identify the shape of the level curves, we must complete the square for xx and yy. 9(x2−2x)+4(y2+4y)=k9(x^2 - 2x) + 4(y^2 + 4y) = k 9(x2−2x+1)+4(y2+4y+4)=k+9(1)+4(4)9(x^2 - 2x + 1) + 4(y^2 + 4y + 4) = k + 9(1) + 4(4) 9(x−1)2+4(y+2)2=k+259(x-1)^2 + 4(y+2)^2 = k + 25 For any k>−25k > -25, this equation describes an ellipse centered at (1,−2)(1, -2). A surface whose level curves are ellipses is an elliptic paraboloid. Since the coefficients of the squared terms (9 and 4) are both positive, the paraboloid opens upwards, and its vertex represents a local minimum. The vertex occurs at the center of the ellipses, which is (1,−2)(1, -2). Thus, the surface is an elliptic paraboloid with a local minimum at (1,−2)(1, -2).