The line 4x+3y-4=0 divides the circumference of the circle centred at (5,3) in the ratio 1:2. Then the equation of the circle is

Options

A. `x^2+y^2-10x-6y-66=0`
B. `x^2+y^2-10x-6y+100=0`
C. `x^2+y^2-10x-6y+66=0`
D. none of these

Answer

To find the equation of the circle centered at (5, 3) that is divided by the line \(4x + 3y - 4 = 0\) in the ratio \(1:2\), we can follow these steps:

### Step 1: Understand the Geometry
The line divides the circumference of the circle into two arcs, one arc being one-third of the circle and the other arc being two-thirds. This means that the angle subtended by the line at the center of the circle is \(120^\circ\) (since \(360^\circ \times \frac{1}{3} = 120^\circ\)).

### Step 2: Calculate the Perpendicular Distance from the Center to the Line
We can use the formula for the perpendicular distance \(d\) from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\):
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = 4\), \(B = 3\), \(C = -4\), and the center of the circle is \((x_1, y_1) = (5, 3)\).

Substituting these values into the formula:
\[ d = \frac{|4(5) + 3(3) - 4|}{\sqrt{4^2 + 3^2}} = \frac{|20 + 9 - 4|}{\sqrt{16 + 9}} = \frac{|25|}{\sqrt{25}} = \frac{25}{5} = 5 \]

### Step 3: Determine the Radius of the Circle
Using the angle \(120^\circ\), we can find the radius \(r\) of the circle. The angle subtended at the center is \(120^\circ\), and the half-angle is \(60^\circ\). Using the cosine of the half-angle:
\[ \cos(60^\circ) = \frac{d}{r} \]
Substituting \(d = 5\):
\[ \frac{1}{2} = \frac{5}{r} \implies r = 10 \]

### Step 4: Write the Equation of the Circle
The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = 5\), \(k = 3\), and \(r = 10\):
\[ (x - 5)^2 + (y - 3)^2 = 10^2 \]
\[ (x - 5)^2 + (y - 3)^2 = 100 \]

### Step 5: Expand the Equation
Now, we can expand the equation:
\[ (x^2 - 10x + 25) + (y^2 - 6y + 9) = 100 \]
Combining like terms:
\[ x^2 + y^2 - 10x - 6y + 34 - 100 = 0 \]
\[ x^2 + y^2 - 10x - 6y - 66 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 10x - 6y - 66 = 0 \]

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