Equation of an ellipse with centre at the origin passing through (5, 0) and having eccentricity 2/3 is:

Options

A. `4x^2+9y^2=100`
B. `9x^2 +5y^2=225`
C. `5x^2+9y^2=125`
D. `6x^2+4y^2=150`

Answer

To find the equation of the ellipse with the given conditions, we will follow these steps:

### Step 1: Identify the standard form of the ellipse
The standard form of an ellipse centered at the origin is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

### Step 2: Use the point (5, 0)
Since the ellipse passes through the point (5, 0), we can substitute \(x = 5\) and \(y = 0\) into the equation:
\[ \frac{5^2}{a^2} + \frac{0^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{25}{a^2} = 1 \]
From this, we can solve for \(a^2\):
\[ a^2 = 25 \]

### Step 3: Use the eccentricity
The eccentricity \(e\) of an ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Given that the eccentricity \(e = \frac{2}{3}\), we can substitute this into the formula:
\[ \frac{2}{3} = \sqrt{1 - \frac{b^2}{25}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{2}{3}\right)^2 = 1 - \frac{b^2}{25} \]
This simplifies to:
\[ \frac{4}{9} = 1 - \frac{b^2}{25} \]

### Step 5: Rearrange to solve for \(b^2\)
Rearranging the equation:
\[ \frac{b^2}{25} = 1 - \frac{4}{9} \]
Calculating the right side:
\[ 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9} \]
Thus:
\[ \frac{b^2}{25} = \frac{5}{9} \]
Multiplying both sides by 25 gives:
\[ b^2 = \frac{125}{9} \]

### Step 6: Write the equation of the ellipse
Now that we have \(a^2\) and \(b^2\):
\[ a^2 = 25 \quad \text{and} \quad b^2 = \frac{125}{9} \]
We can substitute these values into the standard form of the ellipse:
\[ \frac{x^2}{25} + \frac{y^2}{\frac{125}{9}} = 1 \]
To simplify, multiply through by 125:
\[ \frac{125x^2}{25} + 9y^2 = 125 \]
This simplifies to:
\[ 5x^2 + 9y^2 = 125 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ 5x^2 + 9y^2 = 125 \]

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