Problem 51 - Algebra and Trigonometry

Question

Use a graphing utility to graph each equation. $$7 x^{2}+6 x y+2.5 y^{2}-14 x+4 y+9=0$$

Step-by-Step Solution

Answer

The graph of the given equation is an ellipse. This is determined by calculating the discriminant value which is less than 0. The graph is plotted using any graphing utility by entering the given equation. Please check the graphing tool for the graphical representation.

Step 1: Identify the Type of the Graph

The given equation is \(7x^2 + 6xy + 2.5 y^2 - 14x + 4y + 9 = 0\). The general form of a conic section is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), where \(A,B,C,D,E\) and \(F\) are constants, and \(B^2 - 4AC\) helps us determine the type of the section. If \(B^2 - 4AC > 0\), it's a hyperbola. If \(B^2 - 4AC = 0\), it's a parabola, and if \(B^2 - 4AC < 0\), it's an ellipse. If \(A = C\) and \(B = 0\), then it is a circle.

Step 2: Compute the Discriminant

For our equation, \(A=7\), \(B=6\) and \(C=2.5\). Now compute \(B^2 - 4AC = 6^2 - 4*7*2.5 = 36 - 70 = -34 < 0\). Hence, the equation represents an ellipse.

Step 3: Graph the Equation using a Graphing Utility

Use a graphing tool or utility, such as Desmos, GeoGebra or a graphing calculator. Input the equation exactly as it is: \(7x^2 + 6xy + 2.5 y^2 - 14x + 4y + 9 = 0\) and plot the graph. Make sure to adjust the zoom and axes to get the best view of the graph.