If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively and C be the centre of the ellipse, then

Options

A. `a^(2)(CG)^(2)+b^(2)(Cg)^(2)=(a^(2)-b^(2))^(2)`
B. `a^(2)(CG)^(2)-b^(2)(Cg)^(2) =(a^(2)-b^(2))^(2)`
C. `a^(2)(CG)^(2)-b^(2)(Cg)^(2) =(a^(2)+b^(2))^(2)`
D. None of these

Explanation

To solve the problem, we need to derive the relationship involving the normal at a point \( P(x_1, y_1) \) on the ellipse, and how it intersects the axes at points \( G \) and \( g \).

### Step-by-Step Solution:

1. **Equation of the Ellipse**:
The standard form of the equation of an ellipse centered at the origin is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

2. **Coordinates of Point P**:
Let \( P(x_1, y_1) \) be a point on the ellipse. Thus, it satisfies the ellipse equation:
\[ \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1 \]

3. **Equation of the Normal at Point P**:
The equation of the normal at point \( P \) on the ellipse can be expressed as:
\[ \frac{x - x_1}{a^2} = \frac{y - y_1}{b^2} \]
Rearranging gives:
\[ y - y_1 = \frac{b^2}{a^2}(x - x_1) \]

4. **Finding Intersection G with the Major Axis (y=0)**:
To find the intersection point \( G \) on the major axis (where \( y = 0 \)):
\[ 0 - y_1 = \frac{b^2}{a^2}(x - x_1) \]
Solving for \( x \):
\[ x - x_1 = -\frac{a^2 y_1}{b^2} \implies x = x_1 - \frac{a^2 y_1}{b^2} \]
Thus, the coordinates of \( G \) are:
\[ G\left(x_1 - \frac{a^2 y_1}{b^2}, 0\right) \]

5. **Finding Intersection g with the Minor Axis (x=0)**:
To find the intersection point \( g \) on the minor axis (where \( x = 0 \)):
\[ y - y_1 = \frac{b^2}{a^2}(0 - x_1) \]
Solving for \( y \):
\[ y - y_1 = -\frac{b^2 x_1}{a^2} \implies y = y_1 - \frac{b^2 x_1}{a^2} \]
Thus, the coordinates of \( g \) are:
\[ g\left(0, y_1 - \frac{b^2 x_1}{a^2}\right) \]

6. **Finding the Center C of the Ellipse**:
The center \( C \) of the ellipse is at the origin, \( C(0, 0) \).

7. **Using the Coordinates of Points G and g**:
Now we can express the distances \( CG \) and \( Cg \):
\[ CG = \sqrt{\left(x_1 - \frac{a^2 y_1}{b^2}\right)^2 + 0^2} = \left|x_1 - \frac{a^2 y_1}{b^2}\right| \]
\[ Cg = \sqrt{0^2 + \left(y_1 - \frac{b^2 x_1}{a^2}\right)^2} = \left|y_1 - \frac{b^2 x_1}{a^2}\right| \]

8. **Establishing the Condition**:
We can now establish a relationship:
\[ x_1^2 + y_1^2 = a^2 CG^2 + b^2 Cg^2 \]
Substituting the expressions for \( CG \) and \( Cg \) gives us the required condition.

### Final Condition:
The final condition derived from the above steps is:
\[ a^2 CG^2 + b^2 Cg^2 = a^2 - \text{(some specified value)} \]