The equation of normal to the ellipse `4x^2 +9y^2 = 72` at point `(3,2)` is:

Options

A. `x=3`
B. `3x-2y=13`
C. `x+y=1`
D. `3x-2y=5`

Answer

To find the equation of the normal to the ellipse \(4x^2 + 9y^2 = 72\) at the point \((3, 2)\), we will follow these steps:

### Step 1: Rewrite the equation of the ellipse in standard form
The given equation of the ellipse is:
\[ 4x^2 + 9y^2 = 72 \]
Dividing through by 72 gives:
\[ \frac{x^2}{18} + \frac{y^2}{8} = 1 \]
This shows that \(A^2 = 18\) and \(B^2 = 8\).

### Step 2: Identify \(A\) and \(B\)
From the standard form, we have:
\[ A = \sqrt{18} = 3\sqrt{2}, \quad B = \sqrt{8} = 2\sqrt{2} \]

### Step 3: Use the formula for the normal to the ellipse
The equation of the normal to the ellipse at the point \((x_1, y_1)\) is given by:
\[ \frac{A^2}{x_1}(x - x_1) - \frac{B^2}{y_1}(y - y_1) = A^2 - B^2 \]
Substituting \(A^2 = 18\), \(B^2 = 8\), \(x_1 = 3\), and \(y_1 = 2\):
\[ \frac{18}{3}(x - 3) - \frac{8}{2}(y - 2) = 18 - 8 \]

### Step 4: Simplify the equation
Calculating each term:
\[ 6(x - 3) - 4(y - 2) = 10 \]
Expanding this gives:
\[ 6x - 18 - 4y + 8 = 10 \]
Combining like terms:
\[ 6x - 4y - 10 = 0 \]

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ 3x - 2y = 5 \]

### Final Answer
Thus, the equation of the normal to the ellipse at the point \((3, 2)\) is:
\[ 3x - 2y = 5 \]