If the tangents AB and AQ are drawn from the point A(3, -1) to the circle `x^(2)+y^(2)-3x+2y-7=0` and C is the centre of circle, then the area of quadrilateral APCQ is :

Options

A. 9
B. 4
C. 2
D. non-existent

Answer

To solve the problem, we need to find the area of the quadrilateral APCQ, where A is the point from which the tangents are drawn to the circle defined by the equation \(x^2 + y^2 - 3x + 2y - 7 = 0\).

### Step 1: Find the center and radius of the circle

The equation of the circle can be rewritten in standard form by completing the square.

1. Rearranging the equation:
\[ x^2 - 3x + y^2 + 2y - 7 = 0 \]

2. Completing the square for \(x\):
\[ x^2 - 3x = (x - \frac{3}{2})^2 - \frac{9}{4} \]

3. Completing the square for \(y\):
\[ y^2 + 2y = (y + 1)^2 - 1 \]

4. Substitute back into the equation:
\[ (x - \frac{3}{2})^2 - \frac{9}{4} + (y + 1)^2 - 1 - 7 = 0 \]
\[ (x - \frac{3}{2})^2 + (y + 1)^2 = \frac{9}{4} + 1 + 7 = \frac{9}{4} + \frac{4}{4} + \frac{28}{4} = \frac{41}{4} \]

Thus, the center \(C\) of the circle is \((\frac{3}{2}, -1)\) and the radius \(r\) is \(\sqrt{\frac{41}{4}} = \frac{\sqrt{41}}{2}\).

### Step 2: Calculate the length of the tangent from point A to the circle

The formula for the length of the tangent from a point \(A(x_1, y_1)\) to a circle centered at \(C(h, k)\) with radius \(r\) is given by:
\[ \text{Length of tangent} = \sqrt{(x_1 - h)^2 + (y_1 - k)^2 - r^2} \]

Here, \(A(3, -1)\), \(C(\frac{3}{2}, -1)\), and \(r = \frac{\sqrt{41}}{2}\).

1. Calculate \(h\) and \(k\):
\[ h = \frac{3}{2}, \quad k = -1 \]

2. Substitute into the formula:
\[ \text{Length of tangent} = \sqrt{(3 - \frac{3}{2})^2 + (-1 + 1)^2 - \left(\frac{\sqrt{41}}{2}\right)^2} \]
\[ = \sqrt{(\frac{3}{2})^2 + 0 - \frac{41}{4}} = \sqrt{\frac{9}{4} - \frac{41}{4}} = \sqrt{\frac{-32}{4}} = \sqrt{-8} \]

Since the length of the tangent is imaginary, it indicates that point A lies inside the circle.

### Step 3: Conclusion about the area of quadrilateral APCQ

Since the point A is inside the circle, no tangents can be drawn from A to the circle. Therefore, the area of quadrilateral APCQ does not exist.

### Final Answer

The area of quadrilateral APCQ is **not defined**.

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