To find the length and the midpoint of the chord given by the equation \(4x - 3y + 5 = 0\) with respect to the circle defined by the equation \(x^2 + y^2 - 2x + 4y - 20 = 0\), we will follow these steps:
### Step 1: Identify the Circle's Center and Radius
The equation of the circle is given as:
\[
x^2 + y^2 - 2x + 4y - 20 = 0
\]
We can rewrite this in standard form by completing the square.
1. Rearranging the equation:
\[
(x^2 - 2x) + (y^2 + 4y) = 20
\]
2. Completing the square:
- For \(x^2 - 2x\), we add and subtract \(1\) (which is \((\frac{-2}{2})^2\)):
\[
(x - 1)^2 - 1
\]
- For \(y^2 + 4y\), we add and subtract \(4\) (which is \((\frac{4}{2})^2\)):
\[
(y + 2)^2 - 4
\]
3. Substitute back into the equation:
\[
(x - 1)^2 - 1 + (y + 2)^2 - 4 = 20
\]
\[
(x - 1)^2 + (y + 2)^2 = 25
\]
From this, we can see that the center \(O\) of the circle is at \((1, -2)\) and the radius \(r\) is:
\[
r = \sqrt{25} = 5
\]
### Step 2: Find the Distance from the Center to the Chord
To find the distance \(OP\) from the center \(O(1, -2)\) to the chord \(4x - 3y + 5 = 0\), we use the formula for the distance from a point to a line:
\[
d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]
where \(A = 4\), \(B = -3\), \(C = 5\), and \((x_1, y_1) = (1, -2)\).
Calculating:
\[
d = \frac{|4(1) - 3(-2) + 5|}{\sqrt{4^2 + (-3)^2}} = \frac{|4 + 6 + 5|}{\sqrt{16 + 9}} = \frac{|15|}{5} = 3
\]
So, \(OP = 3\).
### Step 3: Use Pythagorean Theorem to Find Length of the Chord
In triangle \(OAP\) (where \(A\) and \(B\) are the endpoints of the chord and \(P\) is the midpoint):
\[
OA^2 = OP^2 + AP^2
\]
Substituting the known values:
\[
5^2 = 3^2 + AP^2
\]
\[
25 = 9 + AP^2
\]
\[
AP^2 = 16 \implies AP = 4
\]
Since \(P\) is the midpoint of chord \(AB\), the length of the chord \(AB\) is:
\[
AB = 2 \times AP = 2 \times 4 = 8
\]
### Step 4: Find the Midpoint of the Chord
To find the midpoint \(P(H, K)\), we need two equations:
1. The chord equation \(4H - 3K + 5 = 0\).
2. The slope condition since \(OP\) is perpendicular to \(AB\).
The slope of line \(AB\) from its equation is \(\frac{4}{3}\). The slope of \(OP\) (from \(O(1, -2)\) to \(P(H, K)\)) is:
\[
\text{slope of } OP = \frac{K + 2}{H - 1}
\]
Setting the product of the slopes to \(-1\):
\[
\frac{K + 2}{H - 1} \cdot \frac{4}{3} = -1
\]
This gives:
\[
4(K + 2) + 3(H - 1) = 0 \implies 4K + 8 + 3H - 3 = 0 \implies 4K + 3H + 5 = 0 \quad \text{(Equation 1)}
\]
Now substituting \(H\) and \(K\) into the chord equation:
\[
4H - 3K + 5 = 0 \quad \text{(Equation 2)}
\]
### Step 5: Solve the System of Equations
From Equation 1:
\[
4K + 3H + 5 = 0 \implies 4K = -3H - 5 \implies K = -\frac{3H + 5}{4}
\]
Substituting \(K\) into Equation 2:
\[
4H - 3\left(-\frac{3H + 5}{4}\right) + 5 = 0
\]
Multiplying through by \(4\) to eliminate the fraction:
\[
16H + 9H + 15 + 20 = 0 \implies 25H + 35 = 0 \implies H = -\frac{7}{5}
\]
Substituting \(H\) back to find \(K\):
\[
K = -\frac{3(-\frac{7}{5}) + 5}{4} = -\frac{\frac{21}{5} + 5}{4} = -\frac{\frac{21 + 25}{5}}{4} = -\frac{\frac{46}{5}}{4} = -\frac{46}{20} = -\frac{23}{10} = -\frac{1}{5}
\]
### Final Answer
Thus, the midpoint of the chord is:
\[
\left(-\frac{7}{5}, -\frac{1}{5}\right)
\]
And the length of the chord is \(8\).
