Number of normal(s) that can be drwan through the point `(sqrt(2),0))` to the ellipse `(x^(2))/(2)+(y^(2))/(1)=1` is / are

Options

A. 0
B. 1
C. 2
D. More than 2

Explanation

To find the number of normals that can be drawn through the point \((\sqrt{2}, 0)\) to the ellipse given by the equation \(\frac{x^2}{2} + \frac{y^2}{1} = 1\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the parameters of the ellipse**:
The equation of the ellipse is \(\frac{x^2}{2} + \frac{y^2}{1} = 1\). Here, we can identify:
- \(a^2 = 2\) (so \(a = \sqrt{2}\))
- \(b^2 = 1\) (so \(b = 1\))

2. **Write the equation of the normal to the ellipse**:
The equation of the normal at a point \((x_0, y_0)\) on the ellipse can be expressed as:
\[ y = mx \pm \frac{\sqrt{m^2 a^2 - b^2}}{\sqrt{a^2 + m^2 b^2}} \]
where \(m\) is the slope of the normal.

3. **Substitute the point \((\sqrt{2}, 0)\)** into the normal equation**:
We need to find the values of \(m\) such that the point \((\sqrt{2}, 0)\) lies on the normal line. Thus, we set \(y = 0\) and \(x = \sqrt{2}\):
\[ 0 = m\sqrt{2} \pm \frac{\sqrt{m^2 \cdot 2 - 1}}{\sqrt{2 + m^2}} \]

4. **Rearranging the equation**:
This gives us two cases to consider:
\[ m\sqrt{2} = \pm \frac{\sqrt{m^2 \cdot 2 - 1}}{\sqrt{2 + m^2}} \]

5. **Square both sides to eliminate the square root**:
Squaring both sides leads to:
\[ 2m^2 = \frac{m^2 \cdot 2 - 1}{2 + m^2} \]

6. **Cross-multiply to eliminate the fraction**:
\[ 2m^2(2 + m^2) = 2m^2 - 1 \]
Expanding this gives:
\[ 4m^2 + 2m^4 = 2m^2 - 1 \]

7. **Rearranging the equation**:
Bringing all terms to one side results in:
\[ 2m^4 + 2m^2 + 1 = 0 \]

8. **Factor out common terms**:
The equation can be simplified to:
\[ 2m^4 + 2m^2 + 1 = 0 \]
This is a quadratic in terms of \(m^2\).

9. **Check the discriminant**:
The discriminant of the quadratic \(2m^4 + 2m^2 + 1 = 0\) is given by:
\[ D = b^2 - 4ac = 2^2 - 4 \cdot 2 \cdot 1 = 4 - 8 = -4 \]
Since the discriminant is negative, there are no real solutions for \(m^2\).

10. **Conclusion**:
Since there are no real values for \(m\), it follows that there are no normals that can be drawn through the point \((\sqrt{2}, 0)\) to the ellipse.

### Final Answer:
The number of normals that can be drawn through the point \((\sqrt{2}, 0)\) to the ellipse is **0**.