Question

The equation of the parabola whose focus is $(6, 0)$ and directrix is $x = -6$ is:

• A. $y^2 = 24x$
• B. $y^2 = -24x$
• C. $x^2 = 24y$
• D. $x^2 = -24y$

Answer 1

The Correct Option is A

1. Understand the problem:

We need to find the equation of a parabola given its focus at (6, 0) and directrix \( x = -6 \).

2. Recall the definition of a parabola:

A parabola is the locus of points equidistant from the focus and the directrix.

3. Let \( (x, y) \) be any point on the parabola:

The distance from \( (x, y) \) to the focus (6, 0) is:

\[ \sqrt{(x - 6)^2 + (y - 0)^2} \]

The distance from \( (x, y) \) to the directrix \( x = -6 \) is:

\[ |x + 6| \]

4. Set the distances equal and simplify:

\[ \sqrt{(x - 6)^2 + y^2} = |x + 6| \]

Square both sides to eliminate the square root and absolute value:

\[ (x - 6)^2 + y^2 = (x + 6)^2 \]

Expand both sides:

\[ x^2 - 12x + 36 + y^2 = x^2 + 12x + 36 \]

Simplify by canceling \( x^2 \) and 36 from both sides:

\[ -12x + y^2 = 12x \]

Combine like terms:

\[ y^2 = 24x \]

Correct Answer: (A) \( y^2 = 24x \)

Answer 2

The definition of a parabola is the set of all points equidistant from the focus and the directrix.

Let \( (x, y) \) be a point on the parabola.
The distance from \( (x, y) \) to the focus \( (6, 0) \) is:

\[ \sqrt{(x - 6)^2 + y^2} \]

The distance from \( (x, y) \) to the directrix \( x = -6 \) is:

\[ |x + 6| \]

Since these distances are equal:

\[ \sqrt{(x - 6)^2 + y^2} = |x + 6| \]

Square both sides:

\[ (x - 6)^2 + y^2 = (x + 6)^2 \] \[ x^2 - 12x + 36 + y^2 = x^2 + 12x + 36 \] \[ y^2 = 24x \]

Therefore, the equation of the parabola is \( y^2 = 24x \).