The equation of the normal at the point P (2, 3) on the ellipse `9x^(2) + 16y^(2) = 180`, is

Options

A. `3y = 8x - 10`
B. `3y - 8x + 7 = 0`
C. `8y + 3x + 7 = 0`
D. `3x + 2y + 7 = 0`

Answer

To find the equation of the normal at the point \( P(2, 3) \) on the ellipse given by \( 9x^2 + 16y^2 = 180 \), we can follow these steps:

### Step 1: Differentiate the equation of the ellipse
We start with the equation of the ellipse:
\[ 9x^2 + 16y^2 = 180 \]
We differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(9x^2) + \frac{d}{dx}(16y^2) = \frac{d}{dx}(180) \]
This gives us:
\[ 18x + 32y \frac{dy}{dx} = 0 \]

### Step 2: Solve for \(\frac{dy}{dx}\)
Rearranging the equation to solve for \(\frac{dy}{dx}\):
\[ 32y \frac{dy}{dx} = -18x \]
\[ \frac{dy}{dx} = -\frac{18x}{32y} = -\frac{9x}{16y} \]

### Step 3: Substitute the point \( P(2, 3) \)
Now, we substitute the coordinates of point \( P(2, 3) \) into the derivative to find the slope of the tangent at that point:
\[ \frac{dy}{dx} \bigg|_{(2, 3)} = -\frac{9(2)}{16(3)} = -\frac{18}{48} = -\frac{3}{8} \]

### Step 4: Find the slope of the normal
The slope of the normal line is the negative reciprocal of the slope of the tangent:
\[ \text{slope of normal} = -\frac{1}{\left(-\frac{3}{8}\right)} = \frac{8}{3} \]

### Step 5: Use point-slope form to write the equation of the normal
Using the point-slope form of the equation of a line, we have:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (2, 3) \) and \( m = \frac{8}{3} \):
\[ y - 3 = \frac{8}{3}(x - 2) \]

### Step 6: Simplify the equation
Now, we simplify this equation:
\[ y - 3 = \frac{8}{3}x - \frac{16}{3} \]
Adding 3 (which is \( \frac{9}{3} \)) to both sides:
\[ y = \frac{8}{3}x - \frac{16}{3} + \frac{9}{3} \]
\[ y = \frac{8}{3}x - \frac{7}{3} \]

### Step 7: Rearranging to standard form
To express this in standard form, we can multiply through by 3 to eliminate the fraction:
\[ 3y = 8x - 7 \]
Rearranging gives:
\[ 8x - 3y - 7 = 0 \]

### Final Answer
Thus, the equation of the normal at the point \( P(2, 3) \) on the ellipse is:
\[ 8x - 3y - 7 = 0 \]