6.8 Q-2

Question Statement

Find the coordinates of the new origin $(h, k)$ such that the first degree terms are removed from the transformed equation of each of the following. Also, find the transformed equations.

i. $3x^2 - 2y^2 + 24x + 12y + 24 = 0$

ii. $25x^2 + 9y^2 + 50x - 36y - 164 = 0$

iii. $x^2 - y^2 - 6x + 2y + 7 = 0$

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Background and Explanation

To solve these problems, we need to perform a coordinate transformation to eliminate the first-degree terms (terms involving $x$ and $y$) from each equation. The transformation of coordinates involves shifting the origin of the coordinate system to a new point $(h, k)$. This can be achieved using the equations:

$$X = x + h, \quad Y = y + k$$

Where $X$ and $Y$ are the new coordinates, and $x$ and $y$ are the original coordinates. By substituting these into the original equation, we aim to remove the first-degree terms and find the transformed equation.

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Solution

i. $3x^2 - 2y^2 + 24x + 12y + 24 = 0$

1. Transformation of coordinates: Let the new origin be $(h, k)$. The transformation equations are:

$$X = x + h, \quad Y = y + k$$

2. Substitute into the original equation:

$$3(x + h)^2 - 2(y + k)^2 + 24(x + h) + 12(y + k) + 24 = 0$$

Expanding each term:

$$3(x^2 + 2hx + h^2) - 2(y^2 + 2ky + k^2) + 24x + 24h + 12y + 12k + 24 = 0$$

Simplifying:

$$3x^2 + 6hx + 3h^2 - 2y^2 - 4ky - 2k^2 + 24x + 24h + 12y + 12k + 24 = 0$$

3. Group the terms:

$$3x^2 - 2y^2 + x(6h + 24) + y(-4k + 12) + (3h^2 - 2k^2 + 24h + 12k + 24) = 0$$

4. Eliminate the first degree terms:

For the first degree terms to vanish, the coefficients of $x$ and $y$ must be zero:

$$6h + 24 = 0 \quad \text{and} \quad -4k + 12 = 0$$

Solving these:

$$h = -4 \quad \text{and} \quad k = 3$$

5. Substitute $h = -4$ and $k = 3$ into the equation:

$$3x^2 - 2y^2 + 48 - 18 - 96 + 36 + 24 = 0$$

Simplify:

$$3x^2 - 2y^2 - 6 = 0$$

Thus, the transformed equation is:

$$3x^2 - 2y^2 - 6 = 0$$

ii. $25x^2 + 9y^2 + 50x - 36y - 164 = 0$

1. Transformation of coordinates: Again, let the new origin be $(h, k)$, with transformation equations:

$$X = x + h, \quad Y = y + k$$

2. Substitute into the original equation:

$$25(x + h)^2 + 9(y + k)^2 + 50(x + h) - 36(y + k) - 164 = 0$$

Expanding:

$$25(x^2 + 2hx + h^2) + 9(y^2 + 2ky + k^2) + 50x + 50h - 36y - 36k - 164 = 0$$

Simplifying:

$$25x^2 + 50hx + 25h^2 + 9y^2 + 18ky + 9k^2 + 50x + 50h - 36y - 36k - 164 = 0$$

3. Group the terms:

$$25x^2 + 9y^2 + x(50h + 50) + y(18k - 36) + (25h^2 + 9k^2 + 50h - 36k - 164) = 0$$

4. Eliminate the first degree terms:

For the first degree terms to vanish, the coefficients of $x$ and $y$ must be zero:

$$50h + 50 = 0 \quad \text{and} \quad 18k - 36 = 0$$

Solving these:

$$h = -1 \quad \text{and} \quad k = 2$$

5. Substitute $h = -1$ and $k = 2$ into the equation:

$$25x^2 + 9y^2 + 25 + 36 - 50 - 72 - 164 = 0$$

Simplify:

$$25x^2 + 9y^2 - 225 = 0$$

Thus, the transformed equation is:

$$25x^2 + 9y^2 - 225 = 0$$

iii. $x^2 - y^2 - 6x + 2y + 7 = 0$

1. Transformation of coordinates: Let the new origin be $(h, k)$, with transformation equations:

$$X = x + h, \quad Y = y + k$$

2. Substitute into the original equation:

$$(x + h)^2 - (y + k)^2 - 6(x + h) + 2(y + k) + 7 = 0$$

Expanding:

$$(x^2 + 2hx + h^2) - (y^2 + 2ky + k^2) - 6x - 6h + 2y + 2k + 7 = 0$$

Simplifying:

$$x^2 - y^2 + x(2h - 6) + y(2k - 2) + (h^2 - k^2 - 6h + 2k + 7) = 0$$

3. Group the terms:

$$x^2 - y^2 + x(2h - 6) + y(2k - 2) + (h^2 - k^2 - 6h + 2k + 7) = 0$$

4. Eliminate the first degree terms:

For the first degree terms to vanish, the coefficients of $x$ and $y$ must be zero:

$$2h - 6 = 0 \quad \text{and} \quad 2k - 2 = 0$$

Solving these:

$$h = 3 \quad \text{and} \quad k = 1$$

5. Substitute $h = 3$ and $k = 1$ into the equation:

$$x^2 - y^2 + 9 - 1 - 18 + 2 + 7 = 0$$

Simplify:

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

Thus, the transformed equation is:

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

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Key Formulas or Methods Used

• Coordinate Transformation:

$$X = x + h, \quad Y = y + k$$

• First Degree Terms Removal: The conditions for eliminating first degree terms are:

$$6h + 24 = 0 \quad \text{and} \quad -4k + 12 = 0$$

for each equation.

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Summary of Steps

1. Define the new coordinates: $X = x + h$, $Y = y + k$.
2. Substitute the transformation into the equation.
3. Expand and group the terms.
4. Set the coefficients of first degree terms to zero to solve for $h$ and $k$.
5. Substitute $h$ and $k$ back into the equation to find the transformed equation.