The equation of the ellipse referred to its axes as coordinate axes, which passes through the point (2, 2) and (1, 4) is

Options

A. `4x^(2)+y^(2)=4`
B. `4x^(2)+y^(2)=16`
C. `4x^(2)+y^(2)=20`
D. `4x^(2)+y^(2)=24`

Answer

To find the equation of the ellipse that passes through the points (2, 2) and (1, 4), we will follow these steps:

### Step 1: Write the general equation of the ellipse
The general equation of an ellipse centered at the origin with axes along the coordinate axes is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

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

### Step 3: Substitute the second point (1, 4) into the equation
Next, we substitute the point (1, 4) into the ellipse equation:
\[ \frac{1^2}{a^2} + \frac{4^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{1}{a^2} + \frac{16}{b^2} = 1 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \(\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{4}\)
2. \(\frac{1}{a^2} + \frac{16}{b^2} = 1\)

We can subtract Equation 1 from Equation 2:
\[ \left(\frac{1}{a^2} + \frac{16}{b^2}\right) - \left(\frac{1}{a^2} + \frac{1}{b^2}\right) = 1 - \frac{1}{4} \]
This simplifies to:
\[ \frac{15}{b^2} = \frac{3}{4} \]
Cross-multiplying gives:
\[ 15 \cdot 4 = 3 \cdot b^2 \implies 60 = 3b^2 \implies b^2 = 20 \]

### Step 5: Substitute \(b^2\) back into Equation 1
Now we substitute \(b^2 = 20\) back into Equation 1:
\[ \frac{1}{a^2} + \frac{1}{20} = \frac{1}{4} \]
Rearranging gives:
\[ \frac{1}{a^2} = \frac{1}{4} - \frac{1}{20} \]
Finding a common denominator (20):
\[ \frac{1}{4} = \frac{5}{20} \implies \frac{1}{a^2} = \frac{5}{20} - \frac{1}{20} = \frac{4}{20} = \frac{1}{5} \]
Thus, we have:
\[ a^2 = 5 \]

### Step 6: Write the final equation of the ellipse
Now we have \(a^2 = 5\) and \(b^2 = 20\). The equation of the ellipse is:
\[ \frac{x^2}{5} + \frac{y^2}{20} = 1 \]
Multiplying through by 20 to eliminate the denominators gives:
\[ 4x^2 + y^2 = 20 \]

### Final Answer
The equation of the ellipse is:
\[ 4x^2 + y^2 = 20 \]