The position of the point (1, 2) relative to the ellipse `2x^(2)+7y^(2)=20` is :

Options

A. A. Outside the ellipse
B. B. Inside the ellipse but not at the focus
C. C. One the ellipse
D. D. At the focus

Answer

To determine the position of the point (1, 2) relative to the ellipse given by the equation \(2x^2 + 7y^2 = 20\), we will follow these steps:

### Step 1: Substitute the coordinates of the point into the ellipse equation.
We have the point (1, 2), where \(x = 1\) and \(y = 2\). We will substitute these values into the equation of the ellipse.

\[ 2(1)^2 + 7(2)^2 = 20 \]

### Step 2: Calculate the left-hand side of the equation.
Now, we will calculate the left-hand side:

\[ 2(1^2) + 7(2^2) = 2(1) + 7(4) = 2 + 28 = 30 \]

### Step 3: Compare the result with the right-hand side of the equation.
Now, we compare the result we obtained with the right-hand side of the equation:

\[ 30 \quad \text{(left-hand side)} \quad \text{and} \quad 20 \quad \text{(right-hand side)} \]

### Step 4: Determine the position of the point.
Since \(30 > 20\), we conclude that the point (1, 2) lies outside the ellipse.

### Final Conclusion:
The position of the point (1, 2) relative to the ellipse \(2x^2 + 7y^2 = 20\) is **outside the ellipse**.

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