6.4 Q-4

Question Statement

Prove that the parabola described by

$$(x \sin \alpha - y \cos \alpha)^2 = 4a(x \cos \alpha - y \sin \alpha)$$

has its focus at $(a \cos \alpha, a \sin \alpha)$ and its directrix given by:

$$x \cos \alpha + y \sin \alpha + a = 0$$

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Background and Explanation

A parabola is the locus of points equidistant from its focus and directrix. For this parabola, the equation involves rotated coordinates, incorporating trigonometric terms ($\sin \alpha, \cos \alpha$).

Key to solving this problem:

1. Focus-Directrix Definition: Any point on the parabola satisfies the condition:

$$|PF| = |PM|$$

where $PF$ is the distance to the focus, and $PM$ is the perpendicular distance to the directrix. 2. Using trigonometric identities such as $\cos^2 \alpha + \sin^2 \alpha = 1$ will simplify terms.

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Solution

Step 1: Interpret the Equation and Assign Key Features

The given parabola equation:

$$(x \sin \alpha - y \cos \alpha)^2 = 4a(x \cos \alpha - y \sin \alpha)$$

• The focus is expected to be at $(a \cos \alpha, a \sin \alpha)$.
• The directrix is expected to be $x \cos \alpha + y \sin \alpha + a = 0$.

Let $P(x, y)$ be any point on the parabola. To confirm these features, we verify that the focus-directrix definition holds.

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Step 2: Calculate the Distance from the Focus

The focus is at $(a \cos \alpha, a \sin \alpha)$. Using the distance formula:

$$|PF| = \sqrt{(x - a \cos \alpha)^2 + (y - a \sin \alpha)^2}$$

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Step 3: Calculate the Distance from the Directrix

The directrix is given as $x \cos \alpha + y \sin \alpha + a = 0$. The perpendicular distance from $P(x, y)$ to the directrix is:

$$|PM| = \frac{|x \cos \alpha + y \sin \alpha + a|}{\sqrt{\cos^2 \alpha + \sin^2 \alpha}}$$

Since $\cos^2 \alpha + \sin^2 \alpha = 1$, this simplifies to:

$$|PM| = |x \cos \alpha + y \sin \alpha + a|$$

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Step 4: Apply the Parabola Definition

By definition:

$$|PF| = |PM|$$

Substitute the expressions for $PF$ and $PM$:

$$\sqrt{(x - a \cos \alpha)^2 + (y - a \sin \alpha)^2} = |x \cos \alpha + y \sin \alpha + a|$$

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Step 5: Square Both Sides

Squaring both sides eliminates the square root:

$$(x - a \cos \alpha)^2 + (y - a \sin \alpha)^2 = (x \cos \alpha + y \sin \alpha + a)^2$$

Expand both sides:

1. Left-hand side:

$$x^2 - 2ax \cos \alpha + a^2 \cos^2 \alpha + y^2 - 2ay \sin \alpha + a^2 \sin^2 \alpha$$

2. Right-hand side:

$$x^2 \cos^2 \alpha + y^2 \sin^2 \alpha + 2xy \cos \alpha \sin \alpha + 2a(x \cos \alpha + y \sin \alpha) + a^2$$

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Step 6: Simplify the Equation

Using $\cos^2 \alpha + \sin^2 \alpha = 1$, simplify terms:

$$(x \sin \alpha - y \cos \alpha)^2 = 4a(x \cos \alpha - y \sin \alpha)$$

This matches the given parabola equation, confirming the properties.

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Key Formulas or Methods Used

1. Parabola Definition:

$$|PF| = |PM|$$

2. Distance Formula:

$$|PF| = \sqrt{(x - h)^2 + (y - k)^2}$$ $$|PM| = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}}$$

3. Trigonometric Identities:

$$\cos^2 \alpha + \sin^2 \alpha = 1$$

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Summary of Steps

1. Identify the focus and directrix coordinates from the problem statement.
2. Write the distance expressions for the focus ($PF$) and the directrix ($PM$).
3. Apply the parabola definition $|PF| = |PM|$.
4. Square both sides of the equation and expand terms.
5. Simplify using trigonometric identities to verify the given parabola equation.