6.5 Q-2

Question Statement

Find the center, foci, eccentricity, vertices, and directrices of the ellipses for the given equations.

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Background and Explanation

An ellipse is a conic section defined as the set of points where the sum of distances to two fixed points (foci) is constant. The standard equations for ellipses are:

1. Horizontal Major Axis:
   $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, , a > b$

2. Vertical Major Axis:
   $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1, , a > b$

Key parameters:

• a: Semi-major axis
• b: Semi-minor axis
• c: Distance from center to foci, given by $c^2 = a^2 - b^2$
• Eccentricity: $e = \frac{c}{a}$, where $0 < e < 1$

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Solution

(i) $x^2 + 4y^2 = 16$

Step 1: Rewrite in standard form.
Divide through by 16:

$$\frac{x^2}{16} + \frac{y^2}{4} = 1$$

Step 2: Identify key parameters.

• Center: $(0, 0)$
• $a^2 = 16, , b^2 = 4$, so $a = 4, , b = 2$
• $c^2 = a^2 - b^2 = 16 - 4 = 12$, so $c = 2\sqrt{3}$

Step 3: Determine features.

• Vertices: $( \pm 4, 0 )$
• Co-vertices: $( 0, \pm 2 )$
• Foci: $( \pm 2\sqrt{3}, 0 )$

Graph:
Placeholder for the diagram.

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(ii) $9x^2 + y^2 = 18$

Step 1: Rewrite in standard form.
Divide through by 18:

$$\frac{x^2}{2} + \frac{y^2}{18} = 1$$

Step 2: Identify key parameters.

• Center: $(0, 0)$
• $a^2 = 18, , b^2 = 2$, so $a = 3\sqrt{2}, , b = \sqrt{2}$
• $c^2 = a^2 - b^2 = 18 - 2 = 16$, so $c = 4$

Step 3: Determine features.

• Vertices: $(0, \pm 3\sqrt{2})$
• Co-vertices: $(\pm \sqrt{2}, 0)$
• Foci: $(0, \pm 4)$

Graph:
Placeholder for the diagram.

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(iii) $25x^2 + 9y^2 = 225$

Step 1: Rewrite in standard form.
Divide through by 225:

$$\frac{x^2}{9} + \frac{y^2}{25} = 1$$

Step 2: Identify key parameters.

• Center: $(0, 0)$
• $a^2 = 25, , b^2 = 9$, so $a = 5, , b = 3$
• $c^2 = a^2 - b^2 = 25 - 9 = 16$, so $c = 4$

Step 3: Determine features.

• Vertices: $(0, \pm 5)$
• Co-vertices: $(\pm 3, 0)$
• Foci: $(0, \pm 4)$

Graph:
Placeholder for the diagram.

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(iv) $\frac{(2x-1)^2}{4} + \frac{(y+2)^2}{16} = 1$

Step 1: Identify the form.
Rewrite as:

$$\frac{\left(x - \frac{1}{2}\right)^2}{1} + \frac{(y + 2)^2}{16} = 1$$

Step 2: Identify key parameters.

• Center: $\left(\frac{1}{2}, -2\right)$
• $a^2 = 16, , b^2 = 1$, so $a = 4, , b = 1$
• $c^2 = a^2 - b^2 = 16 - 1 = 15$, so $c = \sqrt{15}$

Step 3: Determine features.

• Vertices: $\left(\frac{1}{2}, -6\right)$ and $\left(\frac{1}{2}, 2\right)$
• Co-vertices: $\left(-\frac{1}{2}, -2\right)$ and $\left(\frac{3}{2}, -2\right)$
• Foci: $\left(\frac{1}{2}, -2 \pm \sqrt{15}\right)$

Graph:
Placeholder for the diagram.

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(v) $x^2 + 16x + 4y^2 - 16y + 76 = 0$

Step 1: Complete the square.
Rewrite as:

$$(x+8)^2 + 4(y-2)^2 = 4$$

Divide by 4:

$$\frac{(x+8)^2}{4} + \frac{(y-2)^2}{1} = 1$$

Step 2: Identify key parameters.

• Center: $(-8, 2)$
• $a^2 = 4, , b^2 = 1$, so $a = 2, , b = 1$
• $c^2 = a^2 - b^2 = 4 - 1 = 3$, so $c = \sqrt{3}$

Step 3: Determine features.

• Vertices: $(-10, 2)$ and $(-6, 2)$
• Co-vertices: $(-8, 1)$ and $(-8, 3)$
• Foci: $(-8 \pm \sqrt{3}, 2)$

Graph:
Placeholder for the diagram.

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Key Formulas or Methods Used

• Standard Form of Ellipse:
  Horizontal:
  $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
  Vertical:
  $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$
• Relation between axes: $c^2 = a^2 - b^2$
• Eccentricity: $e = \frac{c}{a}$

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Summary of Steps

1. Rewrite the given equation in standard form.
2. Identify the center, major/minor axes, and lengths.
3. Calculate $c$ using $c^2 = a^2 - b^2$.
4. Determine vertices, co-vertices, and foci.
5. Sketch the graph based on the calculated parameters.

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