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Chapter 10: Problem 34

Identify each equation without applying a rotation of axes. $$3 x^{2}-2 \sqrt{3} x y+y^{2}+2 x+2 \sqrt{3} y=0$$

Short Answer

Expert verified
The equation is an ellipse with its center shifted, not rotated. The standard form is \(3(x + \frac{1}{3})^2 - 2\sqrt{3}(x + \frac{1}{3})(y + \sqrt{3}) + (y + \sqrt{3})^2 = \frac{10}{9}\).

Step by step solution

01

Regroup the equation.

Rewrite the given equation by grouping the \(x\) and \(y\)-terms together. This results in: \(3x^2 + 2x - 2\sqrt{3}xy + y^2 + 2\sqrt{3}y = 0\)
02

Complete the square for the x-terms.

To complete the square, halve the coefficient of the \(x\)-term, square it and add it to both sides of the equation. For this equation, the \(x\)-terms can be rewritten as: \(3(x^2 + \frac{2}{3}x + \frac{1}{9}) - 2\sqrt{3}xy + y^2 + 2\sqrt{3}y = \frac{1}{9}\)
03

Complete the square for the y-terms.

Using the same process as in Step 2, complete the square for the \(y\)-terms. The equation then becomes: \(3(x + \frac{1}{3})^2 - 2\sqrt{3}xy + (y^2 + 2\sqrt{3}y + 3) = \frac{1}{9} + 3\)
04

Rewrite the equation in standard form.

Once the squares are completed, rewrite the equation to resemble the standard form of an ellipse equation. The final result will be: \(3(x + \frac{1}{3})^2 - 2\sqrt{3}(x + \frac{1}{3})(y + \sqrt{3}) + (y + \sqrt{3})^2 = \frac{10}{9}\)

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Most popular questions from this chapter

An Earth satellite has an elliptical orbit described by $$\frac{x^{2}}{(5000)^{2}}+\frac{y^{2}}{(4750)^{2}}=1$$ (All units are in miles.) The coordinates of the center of Earth are \((16,0)\) a. The perigee of the satellite's orbit is the point that is nearest Earth's center. If the radius of Earth is approximately 4000 miles, find the distance of the perigee above Earth's surface. b. The apogee of the satellite's orbit is the point that is the greatest distance from Earth's center. Find the distance of the apogee above Earth's surface.

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In Exercises 94鈥�97, determine whether each statement is true or false. If the statement is false, make the necessary If the parabola whose equation is \(x=a y^{2}+b y+c\) has its vertex at \((3,2)\) and \(a>0,\) then it has no \(y\) -intercepts.
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