The condition that the chord of the ellipse `(x^2)/(9) + (y^2)/(4) = 1` whose middle point is`(x_1,y_1)` subtends a right angle at the centre of the ellipse is .........,

Answer

To determine the condition that the chord of the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) whose midpoint is \((x_1, y_1)\) subtends a right angle at the center of the ellipse, we can follow these steps:

### Step 1: Understand the Equation of the Ellipse
The given equation of the ellipse is:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
This represents an ellipse centered at the origin \((0, 0)\) with semi-major axis \(a = 3\) (along the x-axis) and semi-minor axis \(b = 2\) (along the y-axis).

### Step 2: Find the Condition for the Chord
Let the endpoints of the chord be \((x_1 - h, y_1 - k)\) and \((x_1 + h, y_1 + k)\). The midpoint of the chord is given as \((x_1, y_1)\).

### Step 3: Use the Property of Right Angles
For the chord to subtend a right angle at the center of the ellipse, the slopes of the lines connecting the center to the endpoints of the chord must satisfy the condition:
\[ m_1 \cdot m_2 = -1 \]
where \(m_1\) and \(m_2\) are the slopes of the lines from the center to the endpoints of the chord.

### Step 4: Calculate the Slopes
The slopes of the lines from the center \((0, 0)\) to the points \((x_1 - h, y_1 - k)\) and \((x_1 + h, y_1 + k)\) are:
\[ m_1 = \frac{y_1 - k}{x_1 - h}, \quad m_2 = \frac{y_1 + k}{x_1 + h} \]

### Step 5: Set Up the Equation
Setting up the equation for the slopes:
\[ \left(\frac{y_1 - k}{x_1 - h}\right) \cdot \left(\frac{y_1 + k}{x_1 + h}\right) = -1 \]

### Step 6: Substitute the Points into the Ellipse Equation
Since both endpoints of the chord lie on the ellipse, they must satisfy the ellipse equation:
\[ \frac{(x_1 - h)^2}{9} + \frac{(y_1 - k)^2}{4} = 1 \]
\[ \frac{(x_1 + h)^2}{9} + \frac{(y_1 + k)^2}{4} = 1 \]

### Step 7: Solve the System of Equations
From the above equations, we can derive the conditions on \(h\) and \(k\) in terms of \(x_1\) and \(y_1\).

### Step 8: Final Condition
After manipulating the equations, we can derive the final condition that relates \(x_1\) and \(y_1\) to the ellipse parameters. The condition that must hold is:
\[ \frac{x_1^2}{9} + \frac{y_1^2}{4} = 1 \]
This indicates that the midpoint \((x_1, y_1)\) must lie on the ellipse itself.