6.8 Q-3

Question Statement

Find the equation of the curve referred to the new axis obtained by rotating the axis about the origin through the given angle for each of the following:

i. $\quad xy = 1$, with $\theta = 45^\circ$

ii. $\quad 7x^2 - 8xy + y^2 - 9 = 0$, with $\theta = \arctan(2)$

iii. $\quad 9x^2 + 12xy + 4y^2 - x - y = 0$, with $\theta = \arctan \left(\frac{2}{3}\right)$

iv. $\quad x^2 - 2xy + y^2 - 2\sqrt{2}x - 2\sqrt{2}y + 2 = 0$, with $\theta = 45^\circ$

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Background and Explanation

To solve these problems, we need to use the coordinate rotation formula to transform the equations of the curves to new coordinates after rotating the axes by a given angle. The general transformation of the coordinates $x$ and $y$ to new coordinates $X$ and $Y$ after rotating by an angle $\theta$ is given by:

$$X = x \cos \theta - y \sin \theta$$ $$Y = x \sin \theta + y \cos \theta$$

Substituting these transformed coordinates into the original equation of the curve will give the new equation in terms of $X$ and $Y$.

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Solution

i. $xy = 1$, with $\theta = 45^\circ$

1. Equation of transformation:

$$X = x \cos 45^\circ - y \sin 45^\circ = \frac{x - y}{\sqrt{2}}, \quad Y = x \sin 45^\circ + y \cos 45^\circ = \frac{x + y}{\sqrt{2}}$$

2. Substitute into the original equation $xy = 1$:

$$\left( \frac{x - y}{\sqrt{2}} \right) \left( \frac{x - y}{\sqrt{2}} \right) = 1$$

3. Simplify:

$$\frac{(x - y)^2}{2} = 1$$ $$(x - y)^2 = 2$$

4. Transformed equation:

$$x^2 - y^2 = 2$$

ii. $7x^2 - 8xy + y^2 - 9 = 0$, with $\theta = \arctan(2)$

1. Given:

$$\tan \theta = 2, \quad \sin \theta = \frac{2}{\sqrt{5}}, \quad \cos \theta = \frac{1}{\sqrt{5}}$$

2. Equation of transformation:

$$X = \frac{x - 2y}{\sqrt{5}}, \quad Y = \frac{2x + y}{\sqrt{5}}$$

3. Substitute into the original equation:

$$7\left(\frac{x - 2y}{\sqrt{5}}\right)^2 - 8\left(\frac{x - 2y}{\sqrt{5}}\right)\left(\frac{2x + y}{\sqrt{5}}\right) + \left(\frac{2x + y}{\sqrt{5}}\right)^2 - 9 = 0$$

4. Simplify: After expanding and combining like terms, we get:

$$-5x^2 + 45y^2 - 45 = 0$$ $$x^2 - 9y^2 + 9 = 0$$

5. Transformed equation:

$$x^2 - 9y^2 + 9 = 0$$

iii. $9x^2 + 12xy + 4y^2 - x - y = 0$, with $\theta = \arctan\left(\frac{2}{3}\right)$

1. Given:

$$\tan \theta = \frac{2}{3}, \quad \sin \theta = \frac{2}{\sqrt{13}}, \quad \cos \theta = \frac{3}{\sqrt{13}}$$

2. Equation of transformation:

$$X = \frac{3x - 2y}{\sqrt{13}}, \quad Y = \frac{2x + 3y}{\sqrt{13}}$$

3. Substitute into the original equation: After substituting and simplifying the equation, we arrive at:

$$169x^2 - 5\sqrt{13}x - \sqrt{13}y = 0$$

4. Transformed equation:

$$13\sqrt{13}x^2 - 5x - y = 0$$

iv. $x^2 - 2xy + y^2 - 2\sqrt{2}x - 2\sqrt{2}y + 2 = 0$, with $\theta = 45^\circ$

1. Equation of transformation:

$$X = \frac{x - y}{\sqrt{2}}, \quad Y = \frac{x + y}{\sqrt{2}}$$

2. Substitute into the original equation: After substituting and simplifying, we get:

$$4y^2 - 8x + 4 = 0$$ $$y^2 - 2x + 1 = 0$$

3. Transformed equation:

$$y^2 - 2x + 1 = 0$$

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Key Formulas or Methods Used

• Coordinate Rotation Transformation:

$$X = x \cos \theta - y \sin \theta, \quad Y = x \sin \theta + y \cos \theta$$

• Substitute into the original equation to derive the transformed equation in $X$ and $Y$.

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Summary of Steps

1. Determine the rotation angle and find $\cos \theta$ and $\sin \theta$.
2. Apply the coordinate transformation equations to convert $x$ and $y$ to $X$ and $Y$.
3. Substitute the transformed coordinates into the original equation.
4. Simplify the resulting expression to obtain the transformed equation.