the equation of the ellipse passing through (2,1) having e=1/2 , is

Options

A. `3x^(2)+4y^(2)=16`
B. `3x^(2)+5y^(2)=17`
C. `5x^(2)+3y^(2)=23`
D. none of these

Answer

To find the equation of the ellipse that passes through the point (2,1) with an eccentricity \( e = \frac{1}{2} \), we can follow these steps:

### Step 1: Write the standard form of the ellipse
Assume the equation of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

### Step 2: Substitute the point (2,1) into the ellipse equation
Since the ellipse passes through the point (2,1), we substitute \( x = 2 \) and \( y = 1 \) into the equation:
\[ \frac{2^2}{a^2} + \frac{1^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{4}{a^2} + \frac{1}{b^2} = 1 \tag{1} \]

### Step 3: Use the eccentricity to find a relationship between \( a \) and \( b \)
The eccentricity \( e \) of the ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Given that \( e = \frac{1}{2} \), we can set up the equation:
\[ \frac{1}{2} = \sqrt{1 - \frac{b^2}{a^2}} \]
Squaring both sides gives:
\[ \frac{1}{4} = 1 - \frac{b^2}{a^2} \]
Rearranging this leads to:
\[ \frac{b^2}{a^2} = 1 - \frac{1}{4} = \frac{3}{4} \]
Thus, we can express \( b^2 \) in terms of \( a^2 \):
\[ b^2 = \frac{3}{4} a^2 \tag{2} \]

### Step 4: Substitute \( b^2 \) from (2) into (1)
Now substitute \( b^2 = \frac{3}{4} a^2 \) into equation (1):
\[ \frac{4}{a^2} + \frac{1}{\frac{3}{4} a^2} = 1 \]
This simplifies to:
\[ \frac{4}{a^2} + \frac{4}{3a^2} = 1 \]
Combining the fractions gives:
\[ \frac{12 + 4}{3a^2} = 1 \]
Thus:
\[ \frac{16}{3a^2} = 1 \]
This leads to:
\[ 3a^2 = 16 \quad \Rightarrow \quad a^2 = \frac{16}{3} \tag{3} \]

### Step 5: Find \( b^2 \) using (2)
Now substitute \( a^2 = \frac{16}{3} \) back into equation (2) to find \( b^2 \):
\[ b^2 = \frac{3}{4} a^2 = \frac{3}{4} \cdot \frac{16}{3} = 4 \tag{4} \]

### Step 6: Write the equation of the ellipse
Now substitute \( a^2 \) and \( b^2 \) into the standard form of the ellipse:
\[ \frac{x^2}{\frac{16}{3}} + \frac{y^2}{4} = 1 \]
To eliminate the fractions, multiply through by 16:
\[ 3x^2 + 4y^2 = 16 \]

### Final Answer
The equation of the ellipse is:
\[ 3x^2 + 4y^2 = 16 \]