How do you find where the line 2x plus y equals 5 intersects with the circle x2 minus 4x plus y2 minus 2y plus 5 equals 9?

Answer 1

Finding the intersect of 2x + y = 5 with the circle x2 - 4x + y2 - 2y + 5 = 9 means finding the values of (x, y) which satisfy both equations

Rearranging the linear equation to make one variable the subject and then substituting this into the equation for the circle means the other variable can be solved and its pair found (by using the linear equation).

2x + y = 5 -> y = 5 - 2x

Substitute for y in the equation of the circle and solve for x:

x2 - 4x + y2 - 2y + 5 = 9

x2 - 4x + (5 - 2x)2 - 2(5 - 2x) + 5 = 9

x2 - 4x + 25 - 20x + 4x2 - 10 + 4x + 5 - 9 = 0

5x2 - 20x + 11 = 0

x = (20 +- sqrt(400-4x5x11))/10

x = (20 +- sqrt(180))/10

x ~= 3.34 or 0.66

Now using the linear equation, work out y:

x ~= 3.34: y ~= 5 - 2x3.34 = -1.68

x ~= 0.66: y ~= 5 - 2x0.66 = 3.68

So the line intersects the circle at (approx):

(3.34, -1.68) and (0.66, 3.68)