Which one of the following points lies outside the ellipse `(x^(2)//a^(2)) + (y^(2)//b^(2))=1 ?`

Options

A. `(a, 0)`
B. `(0, b)`
C. `(-a, 0)`
D. `(a, b)`

Answer

To determine which point lies outside the ellipse given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can follow these steps:

### Step 1: Understand the Condition for Points Relative to the Ellipse
The equation of the ellipse can be rearranged to:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - 1 = 0 \]
If we substitute a point \((x_0, y_0)\) into this equation, we can determine its position relative to the ellipse:
- If \(\frac{x_0^2}{a^2} + \frac{y_0^2}{b^2} < 1\), the point lies inside the ellipse.
- If \(\frac{x_0^2}{a^2} + \frac{y_0^2}{b^2} = 1\), the point lies on the ellipse.
- If \(\frac{x_0^2}{a^2} + \frac{y_0^2}{b^2} > 1\), the point lies outside the ellipse.

### Step 2: Evaluate Each Point
We will evaluate the given points one by one to see which one lies outside the ellipse.

#### Point 1: \((a, 0)\)
Substituting \((a, 0)\):
\[ \frac{a^2}{a^2} + \frac{0^2}{b^2} = 1 + 0 = 1 \]
Thus, \((a, 0)\) lies on the ellipse.

#### Point 2: \((0, b)\)
Substituting \((0, b)\):
\[ \frac{0^2}{a^2} + \frac{b^2}{b^2} = 0 + 1 = 1 \]
Thus, \((0, b)\) lies on the ellipse.

#### Point 3: \((-a, 0)\)
Substituting \((-a, 0)\):
\[ \frac{(-a)^2}{a^2} + \frac{0^2}{b^2} = 1 + 0 = 1 \]
Thus, \((-a, 0)\) lies on the ellipse.

#### Point 4: \((a, b)\)
Substituting \((a, b)\):
\[ \frac{a^2}{a^2} + \frac{b^2}{b^2} = 1 + 1 = 2 \]
Since \(2 > 1\), \((a, b)\) lies outside the ellipse.

### Conclusion
The point that lies outside the ellipse is \((a, b)\).