The equation of the chord of the ellipse `4x^(2) + 9y^(2)= 36` having (3, 2) as mid pt.is

Options

A. 2x+3y=6
B. 2x+3y=12
C. 3x+y=11
D. 3x+2y=13

Answer

To find the equation of the chord of the ellipse \(4x^2 + 9y^2 = 36\) with the midpoint at the point \((3, 2)\), we can follow these steps:

### Step 1: Rewrite the equation of the ellipse in standard form
The given equation of the ellipse is:
\[ 4x^2 + 9y^2 = 36 \]
Dividing the entire equation by 36 gives:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
This shows that \(a^2 = 9\) and \(b^2 = 4\), so \(a = 3\) and \(b = 2\).

### Step 2: Use the formula for the equation of the chord
The formula for the equation of a chord of an ellipse with midpoint \((h, k)\) is given by:
\[ \frac{hx}{a^2} + \frac{ky}{b^2} = 1 \]
Substituting \(h = 3\), \(k = 2\), \(a^2 = 9\), and \(b^2 = 4\):
\[ \frac{3x}{9} + \frac{2y}{4} = 1 \]

### Step 3: Simplify the equation
This simplifies to:
\[ \frac{x}{3} + \frac{y}{2} = 1 \]
To eliminate the fractions, we can multiply through by 6 (the least common multiple of 3 and 2):
\[ 2x + 3y = 6 \]

### Step 4: Rearranging the equation
To express the equation in standard form, we can write it as:
\[ 2x + 3y - 6 = 0 \]
or equivalently,
\[ 2x + 3y = 12 \]

### Final Answer
Thus, the equation of the chord of the ellipse \(4x^2 + 9y^2 = 36\) having \((3, 2)\) as the midpoint is:
\[ 2x + 3y = 12 \]

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