Problem 38 - Precalculus

Question

(a) use the discriminant to classify the graph of the equation, (b) use the Quadratic Formula to solve for \(y\) and (c) use a graphing utility to graph the equation. $$12 x^{2}-6 x y+7 y^{2}-45=0$$

Step-by-Step Solution

Answer

The graph of the equation represents an ellipse. The general form of the solution obtained for \( y \) is \( y = \frac{-6x ± √[36x^2 - 4*7*12x^2 + 1260]}{14} \). The graph can be obtained using any graphing utility by entering the original equation.

Step 1: Classify the graph using the discriminant

A conic section is given by the general equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). It is classified by the discriminant \( B^2 - 4AC \). If \( B^2 - 4AC > 0 \), the equation represents a hyperbola. If \( B^2 - 4AC = 0 \), it represents a parabola. If \( B^2 - 4AC < 0 \), it represents an ellipse. In the given equation, \( A = 12 \), \( B = -6 \) and \( C = 7 \). Substitute those values into the discriminant: \( (-6)^2 - 4*12*7 = 36 - 336 = -300 \). Since \( B^2 - 4AC < 0 \), the graph represents an ellipse.

Step 2: Solve for y using the Quadratic Formula

The quadratic formula is \( y = \frac{-B ± √(B^2 - 4AC)}{2A} \). However, this equation is not in the standard quadratic form. Let's arrange it: \( 7y^2 + 6xy - 12x^2 + 45 = 0 \). Now, let's use the quadratic formula to solve for y. substitute \( a = 7 \), \( b = 6x \), and \( c = -12x^2 + 45 \) into the formula: \( y = \frac{-6x ± √[(6x)^2 - 4*7*(-12x^2+45)]}{2*7} \). This simplifies to \( y = \frac{-6x ± √[36x^2 - 4*7*12x^2 + 1260]}{14} \). The exact solution will vary depending on the value of x, but this is the general form of the solution.

Step 3: Graph the equation using a graphing utility

To graph this equation, one can use any graphing calculator or graphing utility. Simply enter the original equation \( 12x^2 - 6xy + 7y^2 - 45 = 0 \) into the utility. The graph should resemble an ellipse, as classified in step 1, given the discriminant value.