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: Identify the center and the radius of the circle
The center of the circle is given as \(C(5, 3)\). We need to find the radius of the circle.

### Step 2: Calculate the perpendicular distance from the center to the line
To find the radius, we first need to calculate the perpendicular distance from the center of the circle to the line. The formula for the distance \(d\) from a point \((h, k)\) to a line \(Ax + By + C = 0\) is given by:

\[ d = \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} \]

For the line \(4x + 3y - 4 = 0\), we have:
- \(A = 4\)
- \(B = 3\)
- \(C = -4\)

Substituting \(h = 5\) and \(k = 3\):

\[ d = \frac{|4(5) + 3(3) - 4|}{\sqrt{4^2 + 3^2}} = \frac{|20 + 9 - 4|}{\sqrt{16 + 9}} = \frac{|25|}{\sqrt{25}} = \frac{25}{5} = 5 \]

### Step 3: Determine the radius using the ratio of division
Since the line divides the circle in the ratio \(1:2\), we can find the radius \(r\) of the circle. The distance from the center to the chord (the line) is \(5\), and the radius can be calculated using the cosine of the angle that corresponds to the ratio.

Let \(CM\) be the distance from the center to the chord, \(BC\) be the radius, and the angle corresponding to the ratio \(1:2\) gives us \(60^\circ\) (since \(360^\circ\) divided by \(3\) gives \(120^\circ\) for the larger segment, and thus \(60^\circ\) for the smaller segment).

Using the cosine rule:

\[ CM = r \cdot \cos(60^\circ) \implies 5 = r \cdot \frac{1}{2} \implies r = 10 \]

### Step 4: Write the equation of the circle
The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

Substituting \(h = 5\), \(k = 3\), and \(r = 10\):

\[ (x - 5)^2 + (y - 3)^2 = 10^2 \]

This simplifies to:

\[ (x - 5)^2 + (y - 3)^2 = 100 \]

### Step 5: Expand the equation
Expanding the equation gives:

\[ (x^2 - 10x + 25) + (y^2 - 6y + 9) = 100 \]

Combining like terms:

\[ x^2 + y^2 - 10x - 6y + 34 = 100 \]

Rearranging gives:

\[ x^2 + y^2 - 10x - 6y - 66 = 0 \]

### Final Answer
The equation of the circle is:

\[ x^2 + y^2 - 10x - 6y - 66 = 0 \]

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