Let set S consists of all the points (x, y) satisfying `16x^2+25y^2 le 400`. For points in S let maximum and minimum value of `(y-4)/(x-9)` be M and m respectively, then:

Options

A. `M=1`
B. `M=(65)/(7)`
C. `m=1`
D. `m=(7)/(65)`

Answer

To solve the problem, we start with the inequality given for the set \( S \):

\[ 16x^2 + 25y^2 \leq 400 \]

### Step 1: Identify the ellipse

We can rewrite the inequality in standard form by dividing everything by 400:

\[ \frac{x^2}{25} + \frac{y^2}{16} \leq 1 \]

This represents an ellipse centered at the origin (0,0) with semi-major axis \( a = 5 \) (along the x-axis) and semi-minor axis \( b = 4 \) (along the y-axis).

### Step 2: Identify the vertices of the ellipse

The vertices of the ellipse are:

- Major axis endpoints: \( (5, 0) \) and \( (-5, 0) \)
- Minor axis endpoints: \( (0, 4) \) and \( (0, -4) \)

### Step 3: Set up the expression to maximize/minimize

We need to find the maximum and minimum values of the expression:

\[ z = \frac{y - 4}{x - 9} \]

### Step 4: Evaluate the expression at the vertices

We will evaluate \( z \) at the four vertices of the ellipse.

1. **At \( (0, 4) \)**:
\[ z = \frac{4 - 4}{0 - 9} = \frac{0}{-9} = 0 \]

2. **At \( (0, -4) \)**:
\[ z = \frac{-4 - 4}{0 - 9} = \frac{-8}{-9} = \frac{8}{9} \]

3. **At \( (5, 0) \)**:
\[ z = \frac{0 - 4}{5 - 9} = \frac{-4}{-4} = 1 \]

4. **At \( (-5, 0) \)**:
\[ z = \frac{0 - 4}{-5 - 9} = \frac{-4}{-14} = \frac{2}{7} \]

### Step 5: Determine maximum and minimum values

Now we compare the values obtained:

- At \( (0, 4) \): \( z = 0 \)
- At \( (0, -4) \): \( z = \frac{8}{9} \)
- At \( (5, 0) \): \( z = 1 \)
- At \( (-5, 0) \): \( z = \frac{2}{7} \)

From these values, we find:

- Maximum value \( M = 1 \) (at \( (5, 0) \))
- Minimum value \( m = 0 \) (at \( (0, 4) \))

### Conclusion

Thus, the maximum value \( M \) is 1, and the minimum value \( m \) is 0.

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