To find the area of triangle ABC formed by the chord AB of the circle and the point C(7, 1), we will follow these steps:
### Step 1: Find the center and radius of the circle
The equation of the circle is given as:
\[ x^2 + y^2 - 2x + 4y - 20 = 0 \]
We can rewrite it in the standard form:
\[ (x^2 - 2x) + (y^2 + 4y) = 20 \]
Completing the square:
- For \(x^2 - 2x\), we add and subtract \(1\):
\[ (x - 1)^2 - 1 \]
- For \(y^2 + 4y\), we add and subtract \(4\):
\[ (y + 2)^2 - 4 \]
Thus, the equation becomes:
\[ (x - 1)^2 + (y + 2)^2 - 5 = 20 \]
\[ (x - 1)^2 + (y + 2)^2 = 25 \]
From this, we can see that the center of the circle \(O\) is at \((1, -2)\) and the radius \(r\) is:
\[ r = \sqrt{25} = 5 \]
### Step 2: Find the distance from point C(7, 1) to the chord AB
The equation of the chord AB is given as:
\[ 4x - 3y + 5 = 0 \]
To find the perpendicular distance \(CD\) from point \(C(7, 1)\) to the line \(4x - 3y + 5 = 0\), we use the formula for the distance from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\):
\[
\text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]
Here, \(A = 4\), \(B = -3\), \(C = 5\), \(x_1 = 7\), and \(y_1 = 1\):
\[
CD = \frac{|4(7) - 3(1) + 5|}{\sqrt{4^2 + (-3)^2}} = \frac{|28 - 3 + 5|}{\sqrt{16 + 9}} = \frac{|30|}{5} = 6
\]
### Step 3: Find the distance from the center O(1, -2) to the chord AB
Using the same distance formula for point \(O(1, -2)\):
\[
OD = \frac{|4(1) - 3(-2) + 5|}{\sqrt{4^2 + (-3)^2}} = \frac{|4 + 6 + 5|}{5} = \frac{15}{5} = 3
\]
### Step 4: Find the length of the chord AB
Using the Pythagorean theorem in triangle \(OAD\) (where \(D\) is the foot of the perpendicular from \(O\) to \(AB\)):
- \(OA = r = 5\)
- \(OD = 3\)
Using Pythagorean theorem:
\[
AD = \sqrt{OA^2 - OD^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4
\]
Since \(D\) is the midpoint of \(AB\), the length of chord \(AB\) is:
\[
AB = 2 \times AD = 2 \times 4 = 8
\]
### Step 5: Calculate the area of triangle ABC
The area \(A\) of triangle \(ABC\) can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AB \times CD
\]
Substituting the values:
\[
\text{Area} = \frac{1}{2} \times 8 \times 6 = 24
\]
Thus, the area of triangle ABC is \(24\) square units.
