Simultaneous Equations
When A Straight Line Intersects A Circle
Take the following simultaneous equations:
x2 + y2 = 25! !
!
Eq 1
x + y = 1!
!
!
Eq 2
y = 1 - x!
!
!
Eq 2
!
Eq 1
Rearrange Eq 2 in terms of x:
Substitute Eq 2 into Eq 1:
x2 + (1 - x)2 = 25!
Expand out the brackets in Eq 1 and simplify to give a quadratic equation:
x2 + (1 - x)(1 - x) = 25!
Eq 1
x2 + 1 - 2x + x2 = 25!!
Eq 1
2x2 - 2x + 1 = 25!
!
Eq 1
2x2 - 2x - 24 = 0!
!
Eq 1
Solve Eq 1 using the quadratic formula to find the x values of the points of intersection:
ax2 + bx + c = 0
a=2
b = -2
c = -24
x = -b ±√(b2 - 4ac)
2a
x = -(-2) ±√((-2)2 - (4 X 2 X -24))
2X2
x = 2 ±√(4 - (-192))
4
x = 2 ±√(196)
4
x = 2 ± 14
4
x = 4 or -3
Substitute the x values into Eq 2 to find the y values that correspond with the x value solutions:
If x = 4, y = 1 - 4 = -3
If x = -3, y = 1 - (-3) = 4
Solutions are !
and ! !
!
x = 4, y = -3
x = -3, y = 4
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