Find the equation to the circle whose centre is at the point `(alpha , beta)` and which passes through the origin, and prove that the equation of the tangent at the origin is `alphax + beta y = 0`

Answer

To solve the problem, we need to find the equation of the circle with center at the point \((\alpha, \beta)\) that passes through the origin \((0, 0)\). We will also prove that the equation of the tangent at the origin is \(\alpha x + \beta y = 0\).

### Step 1: Write the general equation of a circle
The general equation of a circle with center \((\alpha, \beta)\) and radius \(r\) is given by:
\[ (x - \alpha)^2 + (y - \beta)^2 = r^2 \]

### Step 2: Determine the radius
Since the circle passes through the origin \((0, 0)\), we can substitute these coordinates into the circle's equation:
\[ (0 - \alpha)^2 + (0 - \beta)^2 = r^2 \]
This simplifies to:
\[ \alpha^2 + \beta^2 = r^2 \]

### Step 3: Substitute \(r^2\) back into the circle's equation
Now we can replace \(r^2\) in the circle's equation:
\[ (x - \alpha)^2 + (y - \beta)^2 = \alpha^2 + \beta^2 \]

### Step 4: Expand the equation
Expanding the left-hand side:
\[ (x^2 - 2\alpha x + \alpha^2) + (y^2 - 2\beta y + \beta^2) = \alpha^2 + \beta^2 \]
Combining like terms gives:
\[ x^2 + y^2 - 2\alpha x - 2\beta y + \alpha^2 + \beta^2 = \alpha^2 + \beta^2 \]
Subtracting \(\alpha^2 + \beta^2\) from both sides results in:
\[ x^2 + y^2 - 2\alpha x - 2\beta y = 0 \]

### Step 5: Final equation of the circle
Thus, the equation of the circle is:
\[ x^2 + y^2 - 2\alpha x - 2\beta y = 0 \]

### Step 6: Find the equation of the tangent at the origin
To find the tangent line at the origin, we use the general formula for the tangent to a circle at a point \((x_1, y_1)\):
\[ T = 0 \Rightarrow xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 \]
For the origin \((0, 0)\), this simplifies to:
\[ g \cdot 0 + f \cdot 0 + c = 0 \Rightarrow c = 0 \]
Thus, we have:
\[ \alpha x + \beta y + 0 = 0 \Rightarrow \alpha x + \beta y = 0 \]

### Conclusion
The equation of the circle is:
\[ x^2 + y^2 - 2\alpha x - 2\beta y = 0 \]
And the equation of the tangent at the origin is:
\[ \alpha x + \beta y = 0 \]