Question

compute analytically the equation of the line (m) obtained from rotating the line (m : 5x + 2y = 3) about the point ((-3, -1)). please thoroughly describe your process.

Explanation:

🆕 New Concept Discovered: Rotation of a Line about a Point
Rotating geometric curves using coordinate transformations

To rotate a line \( m \) about a specific center point \( C(x_0, y_0) \) by an angle \( \theta \), we can use coordinate transformations. However, the problem statement does not specify a rotation angle \( \theta \).

In geometry, when a rotation "about a point" is mentioned without a specified angle to find a unique new line, it typically refers to a half-turn (a rotation of \( 180^\circ \) or \( \pi \) radians). A rotation of \( 180^\circ \) about a point \( C(x_0, y_0) \) is equivalent to a point reflection across \( C \).

Let's find the equation of the line \( m' \) obtained by rotating \( m: 5x + 2y = 3 \) by \( 180^\circ \) about the point \( C(-3, -1) \).

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Step 1: Set up the rotation transformation equations

A rotation of \( 180^\circ \) (point reflection) about the center \( C(x_0, y_0) = (-3, -1) \) maps any point \( (x, y) \) on the original line to a new point \( (x', y') \) on the rotated line \( m' \).

The midpoint of the segment connecting \( (x, y) \) and \( (x', y') \) must be the center of rotation \( C \):

$$ \frac{x + x'}{2} = -3 \implies x + x' = -6 $$

$$ \frac{y + y'}{2} = -1 \implies y + y' = -2 $$

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Step 2: Express the original coordinates in terms of the new coordinates

To find the equation of the new line, we solve for the original coordinates \( x \) and \( y \):

$$ x = -6 - x' $$

$$ y = -2 - y' $$

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Step 3: Substitute into the original line equation

Substitute these expressions for \( x \) and \( y \) into the equation of the original line \( m: 5x + 2y = 3 \):

$$ 5(-6 - x') + 2(-2 - y') = 3 $$

Now, expand and simplify the equation:

$$ -30 - 5x' - 4 - 2y' = 3 $$

$$ -5x' - 2y' - 34 = 3 $$

$$ -5x' - 2y' = 37 $$

Multiply the entire equation by \( -1 \) to write it in standard form:

$$ 5x' + 2y' = -37 $$

Replacing \( x' \) and \( y' \) with standard variables \( x \) and \( y \), we get the equation of the rotated line \( m' \).

Answer:

The equation of the rotated line \( m' \) is:

$$ 5x + 2y = -37 $$