To find the equation of the circle with the center at the intersection of the lines [tex]\( x - y = 4 \)[/tex] and [tex]\( 2x + 3y + 7 = 0 \)[/tex], and passing through the point [tex]\((2, 4)\)[/tex], we can follow these steps:
### Step 1: Find the intersection point of the lines
The equations of the lines are:
1. [tex]\( x - y = 4 \)[/tex]
2. [tex]\( 2x + 3y + 7 = 0 \)[/tex]
We need to solve these two equations simultaneously to find their intersection point, i.e., the center of the circle.
First, solve equation (1) for [tex]\( x \)[/tex]:
[tex]\[ x = y + 4 \][/tex]
Substitute this value of [tex]\( x \)[/tex] into equation (2):
[tex]\[ 2(y + 4) + 3y + 7 = 0 \][/tex]
[tex]\[ 2y + 8 + 3y + 7 = 0 \][/tex]
[tex]\[ 5y + 15 = 0 \][/tex]
[tex]\[ 5y = -15 \][/tex]
[tex]\[ y = -3 \][/tex]
Now substitute [tex]\( y = -3 \)[/tex] back into the expression [tex]\( x = y + 4 \)[/tex]:
[tex]\[ x = -3 + 4 \][/tex]
[tex]\[ x = 1 \][/tex]
So, the intersection point (and the center of the circle) is [tex]\( (1, -3) \)[/tex].
### Step 2: Calculate the radius of the circle
The circle passes through the point [tex]\( (2, 4) \)[/tex], so we use the distance formula to find the radius:
[tex]\[ \text{Radius} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the center of the circle [tex]\((1, -3)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is the point [tex]\((2, 4)\)[/tex]:
[tex]\[ \text{Radius} = \sqrt{(1 - 2)^2 + (-3 - 4)^2} \][/tex]
[tex]\[ \text{Radius} = \sqrt{(-1)^2 + (-7)^2} \][/tex]
[tex]\[ \text{Radius} = \sqrt{1 + 49} \][/tex]
[tex]\[ \text{Radius} = \sqrt{50} \][/tex]
[tex]\[ \text{Radius} = 5\sqrt{2} \][/tex]
### Step 3: Write the equation of the circle
The standard form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
For our circle, the center [tex]\((h, k)\)[/tex] is [tex]\((1, -3)\)[/tex] and the radius [tex]\(r\)[/tex] is [tex]\(5\sqrt{2}\)[/tex]. So, the equation of the circle is:
[tex]\[ (x - 1)^2 + (y + 3)^2 = (5\sqrt{2})^2 \][/tex]
[tex]\[ (x - 1)^2 + (y + 3)^2 = 50 \][/tex]
Thus, the equation of the circle is:
[tex]\[ (x - 1)^2 + (y + 3)^2 = 50 \][/tex]
