Find the co-ordinates of the centre of the circle `x^(2) + y^(2) - 4x + 6y = 3` Given that the point A, outside the circle, has co-ordinates (a, b) where a and b are both positive, and that the tangents drawn from A to the circle are parallel to the two axes respectively, find the values of a and b

Answer

To solve the problem step by step, we will first find the coordinates of the center of the circle given by the equation \(x^2 + y^2 - 4x + 6y = 3\). Then, we will determine the coordinates of point A, from which tangents to the circle are drawn parallel to the axes.

### Step 1: Rewrite the Circle Equation
We start with the equation of the circle:
\[ x^2 + y^2 - 4x + 6y = 3 \]
We will rearrange this equation to find the center and radius of the circle.

### Step 2: Complete the Square for x and y
1. For the \(x\) terms: \(x^2 - 4x\)
- To complete the square, take half of \(-4\) (which is \(-2\)), square it (getting \(4\)), and add it inside the equation:
\[ x^2 - 4x = (x - 2)^2 - 4 \]

2. For the \(y\) terms: \(y^2 + 6y\)
- Take half of \(6\) (which is \(3\)), square it (getting \(9\)), and add it inside the equation:
\[ y^2 + 6y = (y + 3)^2 - 9 \]

### Step 3: Substitute Back into the Equation
Substituting these completed squares back into the original equation gives:
\[ (x - 2)^2 - 4 + (y + 3)^2 - 9 = 3 \]
Simplifying this:
\[ (x - 2)^2 + (y + 3)^2 - 13 = 3 \]
\[ (x - 2)^2 + (y + 3)^2 = 16 \]

### Step 4: Identify the Center and Radius
From the equation \((x - 2)^2 + (y + 3)^2 = 4^2\), we can identify:
- The center of the circle is \((2, -3)\).
- The radius of the circle is \(4\).

### Step 5: Determine the Coordinates of Point A
Given that point A has coordinates \((a, b)\) where both \(a\) and \(b\) are positive, and the tangents from A to the circle are parallel to the axes, we can conclude:
- The tangent parallel to the x-axis means that the y-coordinate of point A must be equal to the y-coordinate of the point of tangency on the circle.
- The tangent parallel to the y-axis means that the x-coordinate of point A must be equal to the x-coordinate of the point of tangency on the circle.

### Step 6: Find the Coordinates of Point A
1. The distance from the center of the circle to point A must equal the radius:
- The y-coordinate of point A must be \(b = -3 + 4 = 1\) (since the radius is \(4\) units above the center).
- The x-coordinate of point A must be \(a = 2 + 4 = 6\) (since the radius is \(4\) units to the right of the center).

Thus, the coordinates of point A are:
\[ (a, b) = (6, 1) \]

### Final Answer
The coordinates of point A are \((6, 1)\).