6.4 Q-2

Question Statement

Write an equation for the parabola given the specified elements, including the focus and directrix. Sketch the graph based on the derived equation.

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Background and Explanation

To derive the equation of a parabola, we use its definition: the set of all points equidistant from the focus and the directrix. The general forms for a parabola’s equation vary based on its orientation:

1. Horizontal parabola (opens left/right):

$$y^2 = 4ax$$

2. Vertical parabola (opens up/down):

$$x^2 = 4ay$$

By applying the definition of a parabola ($|PF| = |PM|$, where $F$ is the focus and $M$ is the projection on the directrix), we derive the required equation.

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Solution

i. Focus $(-3, 0)$, Directrix $x = 3$

1. Let $P(x, y)$ be any point on the parabola. Using the definition:

$$|PF| = |PM|$$

Substitute:

$$|PF| = \sqrt{(x + 3)^2 + y^2}, \quad |PM| = |x - 3|$$

Equating:

$$\sqrt{(x + 3)^2 + y^2} = |x - 3|$$

2. Square both sides:

$$(x + 3)^2 + y^2 = (x - 3)^2$$

Simplify:

$$x^2 + 6x + 9 + y^2 = x^2 - 6x + 9$$

3. Rearrange:

$$y^2 = -12x$$

4. The axis of symmetry is $y = 0$ (x-axis). The focus is $(-3, 0)$, and the vertex is at the origin $(0, 0)$. The directrix is $x = 3$.

Diagram Placeholder

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ii. Focus $(2, 5)$, Directrix $y = 1$

1. Let $P(x, y)$ be any point on the parabola. By definition:

$$|PF| = |PM|$$

Substitute:

$$|PF| = \sqrt{(x - 2)^2 + (y - 5)^2}, \quad |PM| = |y - 1|$$

Equating:

$$\sqrt{(x - 2)^2 + (y - 5)^2} = |y - 1|$$

2. Square both sides:

$$(x - 2)^2 + (y - 5)^2 = (y - 1)^2$$

Expand:

$$x^2 - 4x + 4 + y^2 - 10y + 25 = y^2 - 2y + 1$$

Simplify:

$$x^2 - 4x + 28 = 8y$$

3. Rearrange:

$$(x - 2)^2 = 8(y - 3)$$

4. The axis of symmetry is $x = 2$. The focus is $(2, 5)$, and the vertex is at $(2, 3)$. The directrix is $y = 1$.

Diagram Placeholder

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iii. Focus $(-3, 1)$, Directrix $x - 2y - 3 = 0$

1. Let $P(x, y)$ be any point on the parabola. By definition:

$$|PF| = |PM|$$

Substitute:

$$|PF| = \sqrt{(x + 3)^2 + (y - 1)^2}, \quad |PM| = \frac{|x - 2y - 3|}{\sqrt{1^2 + (-2)^2}}$$

Equating:

$$\sqrt{(x + 3)^2 + (y - 1)^2} = \frac{|x - 2y - 3|}{\sqrt{5}}$$

2. Square both sides:

$$5[(x + 3)^2 + (y - 1)^2] = (x - 2y - 3)^2$$

3. Expand and simplify:

$$4x^2 + 4xy + 36x + y^2 - 22y + 41 = 0$$

Diagram Placeholder

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Key Formulas or Methods Used

1. General Equation of Parabola:

   • Horizontal: $y^2 = 4ax$
   • Vertical: $x^2 = 4ay$
   • Transformed (e.g., shifted vertex): $(x - h)^2 = 4a(y - k)$

2. Focus-Directrix Property:

$$|PF| = |PM|$$

3. Distance Formula:

$$|PF| = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$

4. Directrix Distance:

$$|PM| = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}}$$

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Summary of Steps

1. Identify the focus and directrix.
2. Apply the parabola definition: $|PF| = |PM|$.
3. Write the distance formulas for both the focus and directrix.
4. Square both sides and simplify.
5. Rearrange the equation into the standard form.
6. Identify the vertex, axis of symmetry, and other properties.
7. Sketch the graph based on the derived equation.