The number of maximum normals that can be drawn from any point to an ellipse `x^2/a^2+y^2/b^2=1` , is

Options

A. 2
B. 3
C. 4
D. 1

Answer

To solve the problem of finding the maximum number of normals that can be drawn from any point to the ellipse given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we can follow these steps:

### Step 1: Understanding the Geometry of the Ellipse
An ellipse is a closed curve that is symmetric about its axes. The normals to the ellipse can be drawn from any external point, and we need to determine how many such normals can be drawn.

**Hint:** Visualize the ellipse and the point from which you are drawing the normals.

### Step 2: Analyzing the Normals from a Point
From any external point \( P \), we can draw tangents to the ellipse. Each tangent line will have a corresponding normal line that is perpendicular to it. The key observation here is that for each tangent, there is a unique normal.

**Hint:** Remember that each tangent to the ellipse can be associated with a normal line that intersects the ellipse.

### Step 3: Determining the Number of Tangents
For an ellipse, it is known that from any external point, there can be a maximum of four tangents that can be drawn. This is a fundamental property of conic sections.

**Hint:** Consider the relationship between tangents and normals. Each tangent corresponds to one normal.

### Step 4: Conclusion on the Number of Normals
Since each of the four tangents can be associated with a unique normal, the maximum number of normals that can be drawn from any point to the ellipse is also four.

**Hint:** Summarize the findings: if there are four tangents, then there are four normals.

### Final Answer
The maximum number of normals that can be drawn from any point to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is **4**.