The coordinates of the point on the ellipse `16 x ^(2) + 9y ^(2) = 400` where the ordinate decreases at the same rate at which abscissa increases are :

Options

A. `(3, (16)/(3))`
B. `(3, - (16)/(3))`
C. `(-3,(16)/(3))`
D. (3,4)

Answer

To find the coordinates of the point on the ellipse \(16x^2 + 9y^2 = 400\) where the ordinate decreases at the same rate at which the abscissa increases, we can follow these steps:

### Step 1: Understand the relationship between the rates of change
Given that the ordinate (y-coordinate) decreases at the same rate as the abscissa (x-coordinate) increases, we can express this relationship mathematically as:
\[ \frac{dx}{dt} = -\frac{dy}{dt} \]

### Step 2: Differentiate the equation of the ellipse
We start with the equation of the ellipse:
\[ 16x^2 + 9y^2 = 400 \]
Differentiating both sides 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: Substitute the relationship into the differentiated equation
Substituting \(\frac{dy}{dt} = -\frac{dx}{dt}\) into the differentiated equation:
\[ 32x \frac{dx}{dt} + 18y \left(-\frac{dx}{dt}\right) = 0 \]
This simplifies to:
\[ 32x \frac{dx}{dt} - 18y \frac{dx}{dt} = 0 \]
Factoring out \(\frac{dx}{dt}\):
\[ \frac{dx}{dt}(32x - 18y) = 0 \]

### Step 4: Solve for \(x\) and \(y\)
Since \(\frac{dx}{dt} \neq 0\), we can set the expression in the parentheses to zero:
\[ 32x - 18y = 0 \]
Rearranging gives:
\[ 32x = 18y \quad \Rightarrow \quad \frac{x}{y} = \frac{18}{32} = \frac{9}{16} \]
Thus, we have:
\[ x = \frac{9}{16}y \]

### Step 5: Substitute \(x\) back into the ellipse equation
Now, substitute \(x = \frac{9}{16}y\) into the original ellipse equation:
\[ 16\left(\frac{9}{16}y\right)^2 + 9y^2 = 400 \]
This simplifies to:
\[ 16 \cdot \frac{81}{256}y^2 + 9y^2 = 400 \]
\[ \frac{1296}{256}y^2 + 9y^2 = 400 \]
Converting \(9y^2\) to a common denominator:
\[ \frac{1296}{256}y^2 + \frac{2304}{256}y^2 = 400 \]
\[ \frac{3600}{256}y^2 = 400 \]

### Step 6: Solve for \(y^2\)
Multiplying both sides by \(256\):
\[ 3600y^2 = 102400 \]
Dividing by \(3600\):
\[ y^2 = \frac{102400}{3600} = \frac{256}{9} \]
Taking the square root:
\[ y = \pm \frac{16}{3} \]

### Step 7: Find corresponding \(x\) values
Using \(y = \frac{16}{3}\) to find \(x\):
\[ x = \frac{9}{16} \cdot \frac{16}{3} = 3 \]
Thus, the coordinates are:
\[ (3, \frac{16}{3}) \quad \text{and} \quad (-3, -\frac{16}{3}) \]

### Final Answer
The coordinates of the point on the ellipse are:
\[ (3, \frac{16}{3}) \]