Number of points on the ellipse `x^2/a^2+y^2/b^2=1` at which the normal to the ellipse passes through at least one of the foci of the ellipse is

Answer

To find the number of points on the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) at which the normal to the ellipse passes through at least one of the foci, we can follow these steps:

### Step 1: Understand the Geometry of the Ellipse
The ellipse has two foci located at \( (c, 0) \) and \( (-c, 0) \), where \( c = \sqrt{a^2 - b^2} \).

**Hint:** Remember that the foci are determined by the relationship \( c = \sqrt{a^2 - b^2} \).

### Step 2: Determine the Equation of the Normal
For a point \( P(x_0, y_0) \) on the ellipse, the equation of the normal can be derived. The slope of the tangent at \( P \) is given by:
\[ \frac{dy}{dx} = -\frac{b^2 x_0}{a^2 y_0} \]
Thus, the slope of the normal is the negative reciprocal:
\[ m = \frac{a^2 y_0}{b^2 x_0} \]
The equation of the normal line at point \( P(x_0, y_0) \) is:
\[ y - y_0 = \frac{a^2 y_0}{b^2 x_0} (x - x_0) \]

**Hint:** Make sure to apply the point-slope form of the line correctly.

### Step 3: Condition for the Normal to Pass Through the Foci
To find the points where the normal passes through at least one of the foci, substitute the coordinates of the foci into the normal equation.

1. For the focus \( (c, 0) \):
\[ 0 - y_0 = \frac{a^2 y_0}{b^2 x_0} (c - x_0) \]
Rearranging gives:
\[ y_0 \left( b^2 x_0 + a^2 \right) = a^2 c \]

2. For the focus \( (-c, 0) \):
\[ 0 - y_0 = \frac{a^2 y_0}{b^2 x_0} (-c - x_0) \]
Rearranging gives:
\[ y_0 \left( b^2 x_0 - a^2 \right) = -a^2 c \]

**Hint:** Ensure that you set up the equations correctly for both foci.

### Step 4: Analyze the Conditions
From the equations derived, we can analyze the conditions for \( y_0 \) based on the values of \( x_0 \).

1. If \( y_0 = 0 \), then the points \( (a, 0) \) and \( (-a, 0) \) are solutions.
2. For other values of \( y_0 \), we need to solve the equations to find other potential points.

**Hint:** Consider the symmetry of the ellipse and the nature of the normal lines.

### Step 5: Count the Points
From the analysis, we find that:
- There are two points on the ellipse where the normal passes through the foci directly, which are \( (a, 0) \) and \( (-a, 0) \).
- Additionally, due to the symmetry of the ellipse, there will be other points where the normal lines can pass through the foci.

After careful consideration, we conclude that there are a total of **4 points** on the ellipse where the normal passes through at least one of the foci.

**Final Answer:** The number of points on the ellipse where the normal passes through at least one of the foci is **4**.