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Chapter 10: Problem 50

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

Short Answer

Expert verified
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 by step solution

01

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.\
02

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)\
03

Simplify the equation

By simplifying the equation, you will obtain: \((x + 5\sqrt{5})^2 + 4(y + x/2)^2 = 9 + 5*25 = 134.\
04

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).\

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