If C is the centre of the ellipse `9x^(2) + 16y^(2) = 144` and S is one focus. The ratio of CS to major axis, is

Options

A. `sqrt7 : 16`
B. `sqrt7 : 4`
C. `sqrt5 : sqrt7`
D. none of these

Answer

To solve the problem, we need to find the ratio of the distance from the center of the ellipse to one of its foci (CS) to the length of the major axis. Let's go through the solution step by step.

### Step 1: Write the equation of the ellipse in standard form
The given equation of the ellipse is:
\[ 9x^2 + 16y^2 = 144 \]

To convert it into standard form, we divide the entire equation by 144:
\[ \frac{9x^2}{144} + \frac{16y^2}{144} = 1 \]
This simplifies to:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 16\) which gives \(a = 4\)
- \(b^2 = 9\) which gives \(b = 3\)

### Step 3: Determine the coordinates of the foci
Since \(a^2 > b^2\), the major axis is along the x-axis. The coordinates of the foci are given by:
\[ (\pm ae, 0) \]
where \(e\) is the eccentricity.

### Step 4: Calculate the eccentricity \(e\)
The eccentricity \(e\) is calculated using the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values:
\[ e = \sqrt{1 - \frac{9}{16}} = \sqrt{\frac{16 - 9}{16}} = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4} \]

### Step 5: Find the coordinates of the focus \(S\)
Now we can find the coordinates of one focus \(S\):
\[ S = (ae, 0) = \left(4 \cdot \frac{\sqrt{7}}{4}, 0\right) = (\sqrt{7}, 0) \]

### Step 6: Calculate the distance \(CS\)
The center \(C\) of the ellipse is at the origin \((0, 0)\). The distance \(CS\) is simply the x-coordinate of the focus:
\[ CS = \sqrt{7} \]

### Step 7: Calculate the length of the major axis
The length of the major axis is given by:
\[ 2a = 2 \cdot 4 = 8 \]

### Step 8: Find the ratio of \(CS\) to the major axis
Now, we can find the ratio of \(CS\) to the length of the major axis:
\[ \text{Ratio} = \frac{CS}{\text{Length of major axis}} = \frac{\sqrt{7}}{8} \]

### Final Answer
The ratio of \(CS\) to the major axis is:
\[ \frac{\sqrt{7}}{8} \]

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