Find the equation of the ellipse which passes through the points (3,1) and (2,2).

Answer

To find the equation of the ellipse that passes through the points (3, 1) and (2, 2), we will follow these steps:

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

### Step 2: Substitute the first point (3, 1) into the ellipse equation
Substituting the point (3, 1) into the equation:
\[ \frac{3^2}{a^2} + \frac{1^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{9}{a^2} + \frac{1}{b^2} = 1 \quad \text{(Equation 1)} \]

### Step 3: Substitute the second point (2, 2) into the ellipse equation
Now, substituting the point (2, 2) into the equation:
\[ \frac{2^2}{a^2} + \frac{2^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{4}{a^2} + \frac{4}{b^2} = 1 \]
Dividing the entire equation by 4 gives:
\[ \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{4} \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \(\frac{9}{a^2} + \frac{1}{b^2} = 1\)
2. \(\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{4}\)

We can express \(\frac{1}{b^2}\) from Equation 2:
\[ \frac{1}{b^2} = \frac{1}{4} - \frac{1}{a^2} \]

### Step 5: Substitute \(\frac{1}{b^2}\) into Equation 1
Substituting \(\frac{1}{b^2}\) into Equation 1:
\[ \frac{9}{a^2} + \left(\frac{1}{4} - \frac{1}{a^2}\right) = 1 \]
This simplifies to:
\[ \frac{9}{a^2} - \frac{1}{a^2} + \frac{1}{4} = 1 \]
Combining terms gives:
\[ \frac{8}{a^2} + \frac{1}{4} = 1 \]
Subtracting \(\frac{1}{4}\) from both sides:
\[ \frac{8}{a^2} = 1 - \frac{1}{4} = \frac{3}{4} \]
Cross-multiplying gives:
\[ 8 \cdot 4 = 3a^2 \implies 32 = 3a^2 \implies a^2 = \frac{32}{3} \]

### Step 6: Find \(b^2\) using Equation 2
Now substituting \(a^2\) back into Equation 2:
\[ \frac{1}{\frac{32}{3}} + \frac{1}{b^2} = \frac{1}{4} \]
This simplifies to:
\[ \frac{3}{32} + \frac{1}{b^2} = \frac{1}{4} \]
Subtracting \(\frac{3}{32}\) from both sides:
\[ \frac{1}{b^2} = \frac{1}{4} - \frac{3}{32} \]
Finding a common denominator (32):
\[ \frac{1}{b^2} = \frac{8}{32} - \frac{3}{32} = \frac{5}{32} \]
Thus, we have:
\[ b^2 = \frac{32}{5} \]

### Step 7: Write the final equation of the ellipse
Now substituting \(a^2\) and \(b^2\) back into the standard form of the ellipse:
\[ \frac{x^2}{\frac{32}{3}} + \frac{y^2}{\frac{32}{5}} = 1 \]
Multiplying through by 32 gives:
\[ \frac{3x^2}{32} + \frac{5y^2}{32} = 1 \]
Or, multiplying through by 32:
\[ 3x^2 + 5y^2 = 32 \]

### Final Answer:
The equation of the ellipse is:
\[ 3x^2 + 5y^2 = 32 \]