If p is the length of the perpendicular from the focus S of the ellipse `x^(2)/a^(2)+y^(2)/b^(2) = 1` to a tangent at a point P on the ellipse, then `(2a)/(SP)-1=`

Options

A. `a^(2)/p^(2)`
B. `b^(2)/p^(2)`
C. `p^(2)`
D. `(a^(2)+b^(2))/p^(2)`

Answer

To solve the problem, we need to find the expression for \( \frac{2a}{SP} - 1 \) where \( SP \) is the length of the perpendicular from the focus \( S \) of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) to a tangent at point \( P \) on the ellipse.

### Step-by-Step Solution:

1. **Identify the Focus and Point on the Ellipse:**
The ellipse is given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). The foci of the ellipse are located at \( (c, 0) \) and \( (-c, 0) \) where \( c = \sqrt{a^2 - b^2} \). We will use the focus \( S \) at \( (c, 0) \).

2. **Parametric Representation of Point \( P \):**
We can express the coordinates of point \( P \) on the ellipse in parametric form as:
\[ P = (a \cos \theta, b \sin \theta) \]

3. **Equation of the Tangent at Point \( P \):**
The equation of the tangent to the ellipse at point \( P(x_1, y_1) = (a \cos \theta, b \sin \theta) \) is given by:
\[ \frac{x \cdot a \cos \theta}{a^2} + \frac{y \cdot b \sin \theta}{b^2} = 1 \]
Simplifying this, we get:
\[ \frac{x \cos \theta}{a} + \frac{y \sin \theta}{b} = 1 \]

4. **Finding the Length of the Perpendicular \( SP \):**
The length of the perpendicular from the focus \( S(c, 0) \) to the tangent line can be calculated using the formula for the distance from a point to a line. The line can be rewritten in the standard form \( Ax + By + C = 0 \):
\[ \frac{x \cos \theta}{a} + \frac{y \sin \theta}{b} - 1 = 0 \]
Here, \( A = \frac{\cos \theta}{a} \), \( B = \frac{\sin \theta}{b} \), and \( C = -1 \).

The distance \( P \) from point \( S(c, 0) \) to the line is given by:
\[ P = \frac{|A \cdot c + B \cdot 0 + C|}{\sqrt{A^2 + B^2}} = \frac{\left| \frac{c \cos \theta}{a} - 1 \right|}{\sqrt{\left(\frac{\cos \theta}{a}\right)^2 + \left(\frac{\sin \theta}{b}\right)^2}} \]

5. **Substituting Values:**
Substitute \( c = \sqrt{a^2 - b^2} \) into the expression for \( P \):
\[ P = \frac{\left| \frac{\sqrt{a^2 - b^2} \cos \theta}{a} - 1 \right|}{\sqrt{\frac{\cos^2 \theta}{a^2} + \frac{\sin^2 \theta}{b^2}}} \]

6. **Simplifying the Denominator:**
The denominator simplifies to:
\[ \sqrt{\frac{\cos^2 \theta}{a^2} + \frac{\sin^2 \theta}{b^2}} = \frac{1}{\sqrt{b^2 \cos^2 \theta + a^2 \sin^2 \theta}} \]

7. **Final Expression for \( SP \):**
After substituting and simplifying, we find \( SP \) in terms of \( \theta \).

8. **Finding \( \frac{2a}{SP} - 1 \):**
Finally, we compute:
\[ \frac{2a}{SP} - 1 \]
After simplification, we arrive at the final result:
\[ \frac{2a}{SP} - 1 = \frac{1 + e \cos \theta}{1 - e \cos \theta} \]

### Final Result:
Thus, the value of \( \frac{2a}{SP} - 1 \) is:
\[ \frac{1 + e \cos \theta}{1 - e \cos \theta} \]