Please help me to solve this:find the equation to the circle circumscribing the quadrilateral formed by the straight lines, 2x 3y=2, 3x-2y=4, x 2y=3 and 2x-y=3
Some characters are missing. If I assume the missing symbols are + signs, and plot the resulting lines, the picture does not look much like a quadrilateral. Maybe you can fix things a bit?
http://www.wolframalpha.com/input/?i=plot+2x%2B3y%3D2%2C+3x-2y%3D4%2C+x%2B2y%3D3%2C+2x-y%3D3
To find the equation of the circle circumscribing a quadrilateral formed by the given straight lines, we can follow these steps:
Step 1: Find the intersection points of the given lines.
Step 2: Calculate the midpoint of the line segments connecting the intersection points.
Step 3: Find the distance between the midpoint and any of the intersection points.
Step 4: Use the distance as the radius to form the equation of the circle.
Let's go through these steps:
Step 1: Find the intersection points of the given lines.
To find the intersection points, we need to solve the system of equations formed by the lines. We have the following equations:
2x + 3y = 2 ---(1)
3x - 2y = 4 ---(2)
x - 2y = 3 ---(3)
2x - y = 3 ---(4)
By solving equations (1) and (2), we get:
x = 2 and y = -2
By solving equations (1) and (3), we get:
x = -1/7 and y = 11/7
By solving equations (1) and (4), we get:
x = 9/5 and y = 4/5
Step 2: Calculate the midpoint of the line segments connecting the intersection points.
To find the midpoint, we can take the average of the coordinates of any two intersection points.
Taking the midpoint between (2, -2) and (-1/7, 11/7), we get:
Midpoint = ((2+(-1/7))/2, (-2+11/7)/2)
= (13/14, 5/14)
Step 3: Find the distance between the midpoint and any of the intersection points.
Let's find the distance between the midpoint (13/14, 5/14) and one of the intersection points, for example, (2, -2).
Using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((2 - 13/14)^2 + (-2 - 5/14)^2)
= sqrt((28/14 - 13/14)^2 + (-28/14 - 5/14)^2)
= sqrt((15/14)^2 + (-33/14)^2)
= sqrt(225/196 + 1089/196)
= sqrt(1314/196)
= sqrt(6.69)
Step 4: Use the distance as the radius to form the equation of the circle.
Since the distance between the midpoint and any of the intersection points is the radius, we can use it to form the equation of the circle.
Equation of the circle: (x - 13/14)^2 + (y - 5/14)^2 = (6.69)^2
Therefore, the equation of the circle circumscribing the quadrilateral formed by the given lines is:
(x - 13/14)^2 + (y - 5/14)^2 = 44.56