A chord PQ of the ellipse `(x^2)/(9) + (y^2)/(4) = 1` subtends a right angle at the centre of the ellipse. The locus of the point of intersection of the tangents to the ellipse at P and Q is .......

Answer

To find the locus of the point of intersection of the tangents to the ellipse at points P and Q, where the chord PQ subtends a right angle at the center of the ellipse, we can follow these steps:

### Step 1: Write the equation of the ellipse
The given ellipse is:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
Here, \(a^2 = 9\) and \(b^2 = 4\), so \(a = 3\) and \(b = 2\).

### Step 2: Find the chord of contact
The chord of contact for the ellipse at a point \((x_1, y_1)\) is given by the equation:
\[ \frac{xx_1}{9} + \frac{yy_1}{4} = 1 \]

### Step 3: Condition for right angle at the center
Since the chord PQ subtends a right angle at the center of the ellipse, we can use the condition for perpendicular tangents. The condition for two lines to be perpendicular is that the sum of the coefficients of \(x^2\) and \(y^2\) in the equation of the chord of contact must equal zero.

### Step 4: Write the equation in homogeneous form
From the chord of contact, we can write:
\[ \frac{x^2}{9} + \frac{y^2}{4} = \frac{x_1 x}{9} + \frac{y_1 y}{4} \]
Squaring both sides gives us:
\[ \left(\frac{x_1 x}{9} + \frac{y_1 y}{4}\right)^2 = \frac{x^2}{9} + \frac{y^2}{4} \]

### Step 5: Expand and rearrange
Expanding the left-hand side:
\[ \frac{x_1^2 x^2}{81} + 2 \cdot \frac{x_1 y_1}{36} xy + \frac{y_1^2 y^2}{16} = \frac{x^2}{9} + \frac{y^2}{4} \]
Rearranging gives us:
\[ \left(\frac{1}{9} - \frac{x_1^2}{81}\right)x^2 + \left(\frac{1}{4} - \frac{y_1^2}{16}\right)y^2 + \frac{xy}{18} = 0 \]

### Step 6: Apply the perpendicularity condition
For the lines to be perpendicular, we set:
\[ \frac{1}{9} - \frac{x_1^2}{81} + \frac{1}{4} - \frac{y_1^2}{16} = 0 \]

### Step 7: Substitute \(x_1\) and \(y_1\) with \(x\) and \(y\)
Substituting \(x_1\) and \(y_1\) with \(x\) and \(y\) gives:
\[ \frac{x^2}{9} + \frac{y^2}{4} = k \]
where \(k\) is a constant.

### Step 8: Final equation of the locus
After simplifying, we arrive at the locus of the point of intersection of the tangents:
\[ 4x^2 + 9y^2 = 36 \]
or
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

### Conclusion
The locus of the point of intersection of the tangents to the ellipse at points P and Q is another ellipse given by:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]