The point on the ellipse `16x^(2)+9y^(2)=400`, where the ordinate decreases at the same rate at which the abscissa increases is (a, b), then `a+3b` can be

Options

A. 16
B. 19
C. 6
D. 9

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Write the equation of the ellipse
The equation of the ellipse is given as:
\[ 16x^2 + 9y^2 = 400 \]

### Step 2: Differentiate the equation with respect to time \( t \)
We differentiate both sides of the equation with respect to \( t \):
\[ \frac{d}{dt}(16x^2) + \frac{d}{dt}(9y^2) = \frac{d}{dt}(400) \]
This gives us:
\[ 32x \frac{dx}{dt} + 18y \frac{dy}{dt} = 0 \]

### Step 3: Express \( \frac{dy}{dt} \) in terms of \( \frac{dx}{dt} \)
From the problem, we know that the ordinate (y-coordinate) decreases at the same rate the abscissa (x-coordinate) increases, which means:
\[ \frac{dy}{dt} = -\frac{dx}{dt} \]
Substituting this into the differentiated equation:
\[ 32x \frac{dx}{dt} + 18y \left(-\frac{dx}{dt}\right) = 0 \]
This simplifies to:
\[ (32x - 18y) \frac{dx}{dt} = 0 \]
Since \( \frac{dx}{dt} \neq 0 \) (the abscissa is changing), we can set:
\[ 32x - 18y = 0 \]
Thus, we have:
\[ 32x = 18y \quad \Rightarrow \quad y = \frac{16}{9}x \]

### Step 4: Substitute \( y \) back into the ellipse equation
Now, we substitute \( y = \frac{16}{9}x \) into the ellipse equation:
\[ 16x^2 + 9\left(\frac{16}{9}x\right)^2 = 400 \]
This simplifies to:
\[ 16x^2 + 9 \cdot \frac{256}{81}x^2 = 400 \]
\[ 16x^2 + \frac{2304}{81}x^2 = 400 \]
To combine the terms, we need a common denominator:
\[ \frac{1296}{81}x^2 + \frac{2304}{81}x^2 = 400 \]
\[ \frac{3600}{81}x^2 = 400 \]

### Step 5: Solve for \( x^2 \)
Multiplying both sides by \( 81 \):
\[ 3600x^2 = 32400 \]
Dividing by \( 3600 \):
\[ x^2 = 9 \quad \Rightarrow \quad x = 3 \text{ or } x = -3 \]

### Step 6: Find corresponding \( y \) values
Using \( y = \frac{16}{9}x \):
1. For \( x = 3 \):
\[ y = \frac{16}{9} \cdot 3 = \frac{48}{9} = \frac{16}{3} \]
2. For \( x = -3 \):
\[ y = \frac{16}{9} \cdot (-3) = -\frac{16}{3} \]

### Step 7: Identify the points
The points on the ellipse are:
1. \( (3, \frac{16}{3}) \)
2. \( (-3, -\frac{16}{3}) \)

### Step 8: Calculate \( a + 3b \)
For the point \( (3, \frac{16}{3}) \):
\[ a + 3b = 3 + 3 \cdot \frac{16}{3} = 3 + 16 = 19 \]

For the point \( (-3, -\frac{16}{3}) \):
\[ a + 3b = -3 + 3 \cdot \left(-\frac{16}{3}\right) = -3 - 16 = -19 \]

Since we are looking for a positive value, we take:
\[ \text{Final answer: } 19 \]