In the ellipse ` (x^(2))/(36) + (y^(2))/( 9) = 1 ` find the equation to the chord which passes through the point (2,1) and is bisected at that point

Answer

To find the equation of the chord of the ellipse \(\frac{x^2}{36} + \frac{y^2}{9} = 1\) that passes through the point (2, 1) and is bisected at that point, we can follow these steps:

### Step 1: Identify the midpoint of the chord
Since the chord is bisected at the point (2, 1), this point is the midpoint of the chord.

### Step 2: Use the midpoint formula for the chord
The equation of the chord of an ellipse that is bisected at the point \((h, k)\) is given by:
\[ \frac{x_1 x}{a^2} + \frac{y_1 y}{b^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} \]
where \((h, k)\) is the midpoint, and \(a^2\) and \(b^2\) are the denominators of the ellipse equation.

For our ellipse, \(a^2 = 36\) and \(b^2 = 9\). The midpoint is \((h, k) = (2, 1)\).

### Step 3: Substitute the values into the formula
Substituting \(h = 2\) and \(k = 1\) into the formula gives:
\[ \frac{2x}{36} + \frac{1y}{9} = \frac{2^2}{36} + \frac{1^2}{9} \]

### Step 4: Calculate the right side
Calculating the right side:
\[ \frac{4}{36} + \frac{1}{9} = \frac{4}{36} + \frac{4}{36} = \frac{8}{36} = \frac{2}{9} \]

### Step 5: Set up the equation
Now we have:
\[ \frac{2x}{36} + \frac{y}{9} = \frac{2}{9} \]

### Step 6: Clear the denominators
To eliminate the denominators, multiply through by 36:
\[ 2x + 4y = 8 \]

### Step 7: Simplify the equation
Dividing the entire equation by 2 gives:
\[ x + 2y = 4 \]

### Final Answer
Thus, the equation of the chord that passes through the point (2, 1) and is bisected at that point is:
\[ \boxed{x + 2y = 4} \]