The minimum length of intercept on any tangent to the ellipse `(x^(2))/(4)+(y^(2))/(9)=1` cut by the circle `x^(2)+y^(2)=25` is :

Options

A. 8
B. 9
C. 2
D. 11

Answer

To find the minimum length of the intercept on any tangent to the ellipse \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) cut by the circle \(x^2 + y^2 = 25\), we can follow these steps:

### Step 1: Identify the Ellipse and Circle
The given ellipse is \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) and the given circle is \(x^2 + y^2 = 25\).

### Step 2: Determine the Points of Intersection
To find the points where the tangent to the ellipse intersects the circle, we can analyze the ellipse's dimensions. The semi-major axis \(a = 3\) (along the y-axis) and the semi-minor axis \(b = 2\) (along the x-axis).

### Step 3: Find the Tangent Line
The general equation of a tangent to the ellipse at point \((x_0, y_0)\) is given by:
\[ \frac{xx_0}{4} + \frac{yy_0}{9} = 1 \]
We need to find the intercepts of this tangent line with the circle.

### Step 4: Substitute the Tangent Equation into the Circle's Equation
Substituting the tangent line equation into the circle's equation \(x^2 + y^2 = 25\) will allow us to find the points of intersection.

### Step 5: Calculate the Length of the Intercept
The intercept length \(L\) on the tangent line can be calculated using the formula:
\[ L = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the points of intersection.

### Step 6: Minimize the Length
To find the minimum length of the intercept, we can analyze the geometry of the situation. The intercept will be minimized when the tangent line is horizontal or vertical, as these orientations will maximize the distance from the center of the ellipse to the circle.

### Step 7: Calculate Specific Intercepts
For example, if we take the y-coordinate at the top of the ellipse \(y = 3\), we can find the corresponding x-values using the circle's equation:
\[ x^2 + 3^2 = 25 \implies x^2 = 16 \implies x = \pm 4 \]
Thus, the points of intersection are \((-4, 3)\) and \((4, 3)\).

### Step 8: Find the Length of the Intercept
The length of the intercept between these two points is:
\[ L = |x_2 - x_1| = |4 - (-4)| = 4 + 4 = 8 \]

### Conclusion
Therefore, the minimum length of the intercept on any tangent to the ellipse cut by the circle is \(8\).