The maximum distance of the centre of the ellipse `x^2/(81)+y^2/(25)=1` from a normal to the ellipse is

Answer

To find the maximum distance of the center of the ellipse \( \frac{x^2}{81} + \frac{y^2}{25} = 1 \) from a normal to the ellipse, we will follow these steps:

### Step 1: Identify the parameters of the ellipse
The given ellipse can be expressed in the standard form:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a^2 = 81 \) and \( b^2 = 25 \). Thus, we have:
\[ a = \sqrt{81} = 9, \quad b = \sqrt{25} = 5 \]

### Step 2: Write the formula for the distance from the center to the normal
The distance \( S \) from the center of the ellipse (which is at the origin \( (0,0) \)) to the normal at a point on the ellipse is given by:
\[ S = \frac{|a^2 - b^2|}{\sqrt{a^2 \sec^2 \theta + b^2 \cos^2 \theta}} \]
where \( \theta \) is the parameter that defines the point on the ellipse.

### Step 3: Substitute the values of \( a \) and \( b \)
Substituting \( a = 9 \) and \( b = 5 \) into the formula, we have:
\[ S = \frac{|9^2 - 5^2|}{\sqrt{9^2 \sec^2 \theta + 5^2 \cos^2 \theta}} = \frac{|81 - 25|}{\sqrt{81 \sec^2 \theta + 25 \cos^2 \theta}} = \frac{56}{\sqrt{81 \sec^2 \theta + 25 \cos^2 \theta}} \]

### Step 4: Find the maximum value of \( S \)
To maximize \( S \), we need to minimize the denominator \( \sqrt{81 \sec^2 \theta + 25 \cos^2 \theta} \).

We know that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Thus, we can rewrite the denominator:
\[ 81 \sec^2 \theta + 25 \cos^2 \theta = 81(1 + \tan^2 \theta) + 25 \left(\frac{1}{\sec^2 \theta}\right) = 81 + 81 \tan^2 \theta + 25 \cos^2 \theta \]

To minimize this expression, we can use the fact that for \( S \) to be maximum, the denominator must be minimized. The minimum value occurs when \( \tan^2 \theta \) is minimized, which occurs at \( \theta \) such that \( \sec^2 \theta \) is minimized.

### Step 5: Use the relationship for maximum distance
The maximum distance \( S \) can also be directly calculated using the formula:
\[ S_{\text{max}} = |a - b| = |9 - 5| = 4 \]

### Conclusion
Thus, the maximum distance of the center of the ellipse from a normal to the ellipse is:
\[ \boxed{4} \]