The intercept made by the auxiliary circle of the ellipse `(x ^(2))/(a ^(2)) + (y ^(2))/(b ^(2)) =1 (a gt b gt 1)` on any tangent to the ellipse, subtends a right angle at its ceotre if

Options

A. `a le (b)/(2)`
B. `a le (b)/(sqrt2)`
C. `a lt 2b`
D. none of these

Answer

To solve the problem, we need to find the condition under which the intercept made by the auxiliary circle of the ellipse on any tangent to the ellipse subtends a right angle at the center of the ellipse.

### Step-by-Step Solution:

1. **Understanding the Ellipse and Auxiliary Circle**:
The given ellipse is represented by the equation:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a > b > 1 \). The center of this ellipse is at the origin (0, 0).

The auxiliary circle of the ellipse is given by:
\[ x^2 + y^2 = a^2 \]
This circle has the same center as the ellipse.

2. **Equation of the Tangent**:
The equation of the tangent to the ellipse at a point \( (a \cos \theta, b \sin \theta) \) can be expressed as:
\[ \frac{\cos \theta}{a} x + \frac{\sin \theta}{b} y = 1 \]

3. **Finding the Intercepts**:
To find the intercepts on the auxiliary circle, we can substitute \( y = 0 \) to find the x-intercept and \( x = 0 \) to find the y-intercept.

- **X-intercept**:
Set \( y = 0 \):
\[ \frac{\cos \theta}{a} x = 1 \implies x = \frac{a}{\cos \theta} \]

- **Y-intercept**:
Set \( x = 0 \):
\[ \frac{\sin \theta}{b} y = 1 \implies y = \frac{b}{\sin \theta} \]

4. **Finding the Length of the Intercept**:
The length of the intercept made by the tangent on the auxiliary circle is given by the distance between the x-intercept and the y-intercept:
\[ L = \frac{a}{\cos \theta} + \frac{b}{\sin \theta} \]

5. **Condition for Right Angle**:
For the intercept to subtend a right angle at the center, the following condition must hold:
\[ \tan^2 \theta = \frac{b^2}{a^2} \]

6. **Final Condition**:
This condition implies that:
\[ \tan^2 \theta < 1 \quad \text{(since \( b < a \))} \]
Therefore, the condition for the intercept to subtend a right angle at the center of the ellipse is:
\[ \tan^2 \theta = \frac{b^2}{a^2} \quad \text{where } a > b > 1 \]