If the locus of the foot of the perpendicular drawn from centre upon any tangent to the ellipse `(x^(2))/(40)+(y^(2))/(10)=1` is `(x^(2)+y^(2))^(2)=ax^(2)+by^(2)`, then `(a-b)` is equal to

Options

A. 10
B. 20
C. 25
D. 30

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) from the given ellipse and its tangent, and then calculate \( a - b \).

### Step-by-Step Solution:

1. **Identify the Equation of the Ellipse**:
The given ellipse is:
\[ \frac{x^2}{40} + \frac{y^2}{10} = 1 \]
Here, \( a^2 = 40 \) and \( b^2 = 10 \).

2. **Equation of the Tangent to the Ellipse**:
The equation of the tangent to the ellipse at any point can be expressed as:
\[ y = mx + \sqrt{a^2 m^2 + b^2} \]
Substituting \( a^2 \) and \( b^2 \):
\[ y = mx + \sqrt{40m^2 + 10} \]

3. **Finding the Foot of the Perpendicular**:
Since the foot of the perpendicular from the center (0,0) to the tangent line must also lie on the tangent line, we can express the slope of the perpendicular line as:
\[ \text{slope of perpendicular} = -\frac{1}{m} \]
The equation of the perpendicular line passing through the origin is:
\[ y = -\frac{1}{m}x \]

4. **Setting the Two Equations Equal**:
To find the coordinates of the foot of the perpendicular, we set the two equations equal:
\[ mx + \sqrt{40m^2 + 10} = -\frac{1}{m}x \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ mx + \frac{1}{m}x = -\sqrt{40m^2 + 10} \]
Factoring out \( x \):
\[ x(m + \frac{1}{m}) = -\sqrt{40m^2 + 10} \]

6. **Finding \( y \) in Terms of \( x \)**:
Substitute \( x \) back into the line equation to find \( y \):
\[ y = -\frac{1}{m}x \]

7. **Finding the Locus**:
The locus of the foot of the perpendicular can be expressed as:
\[ x^2 + y^2 = k \]
where \( k \) is a constant that we will determine.

8. **Squaring Both Sides**:
From the previous steps, we can express:
\[ (x^2 + y^2)^2 = ax^2 + by^2 \]
By comparing coefficients, we can find \( a \) and \( b \).

9. **Comparing with the Given Form**:
After simplifying, we find:
\[ (x^2 + y^2)^2 = 40x^2 + 10y^2 \]
Thus, \( a = 40 \) and \( b = 10 \).

10. **Calculating \( a - b \)**:
Finally, we calculate:
\[ a - b = 40 - 10 = 30 \]

### Final Answer:
\[ \boxed{30} \]