Problem 50 - Algebra and Trigonometry

Question

Use a graphing utility to graph each equation. $$x^{2}+4 x y+4 y^{2}+10 \sqrt{5} x-9=0$$

Step-by-Step Solution

Answer

The equation after all necessary modifications is \((x + 5\sqrt{5})^2 / 134 + (y + x/2)^2 / 33.5 = 1\). Use a graphing utility to visualize the graph of this ellipse.

Step 1: Simplify the Equation

First group the terms that contain the same variables together. This results in the equation: \((x^2 + 10\sqrt{5}x) + (4xy + 4y^2) - 9 = 0.\

Step 2: Complete the square

As it's a quadratic equation, it's more convenient to convert it to standard form. To do this, complete the square for both x and y related terms. Here, it's important to balance the equation: if you add a term to one side, add the same term on the other side. For the x terms: \((x^2 + 10\sqrt{5}x + (5\sqrt{5})^2)\) and for the y terms: \((4y^2 + 4xy + x^2)\). Adding \((5\sqrt{5})^2\) to the left side of the equation, you should also add it to the right, resulting in the equation: \((x^2 + 10\sqrt{5}x + (5\sqrt{5})^2) + (4y^2 + 4xy + x^2) = 9 + (5\sqrt{5})^2)\

Step 3: Simplify the equation

By simplifying the equation, you will obtain: \((x + 5\sqrt{5})^2 + 4(y + x/2)^2 = 9 + 5*25 = 134.\

Step 4: Divide through by 134

As the standard form of an ellipse equation is \(x^2/a^2 + y^2/b^2 = 1\), divide the equation throughout by 134 to get it in this form: \((x + 5\sqrt{5})^2 / 134 + (y + x/2)^2 / 33.5 = 1).\