The equation of the ellipse passes through the points (4,0) and (0,2) is

Options

A. `x^2/4+y^2/16=1`
B. `x^2/16+y^2/4=1`
C. `x^2/4+y^2/2=1`
D. `x^2/2+y^2/4=1`

Answer

To find the equation of the ellipse that passes through the points (4, 0) and (0, 2), we will start with the general form of the ellipse equation.

### Step-by-Step Solution:

1. **General Form of the Ellipse:**
The standard form of the ellipse centered at the origin is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a^2\) and \(b^2\) are the squares of the semi-major and semi-minor axes respectively.

2. **Substituting the Point (4, 0):**
We will substitute the point (4, 0) into the ellipse equation:
\[ \frac{4^2}{a^2} + \frac{0^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{16}{a^2} = 1 \]
From this, we can find \(a^2\):
\[ a^2 = 16 \]

3. **Substituting the Point (0, 2):**
Next, we substitute the point (0, 2) into the ellipse equation:
\[ \frac{0^2}{a^2} + \frac{2^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{4}{b^2} = 1 \]
From this, we can find \(b^2\):
\[ b^2 = 4 \]

4. **Writing the Equation of the Ellipse:**
Now that we have \(a^2\) and \(b^2\), we can write the equation of the ellipse:
\[ \frac{x^2}{16} + \frac{y^2}{4} = 1 \]

### Final Equation:
Thus, the equation of the ellipse that passes through the points (4, 0) and (0, 2) is:
\[ \frac{x^2}{16} + \frac{y^2}{4} = 1 \]