The graph of 2x − y= 7 is a line. The graph of x^2− 10x + y^2 + 4x = −4 is a circle. The line intersects the circle in two points. Find the coordinates of these two points.
2x − y= 7 ......(1)
x^2− 10x + y^2 + 4x = −4 .....(2)
From 1
y=(2x-7)....(3)
y²+x²-6x=-4
(2x-7)²+x²-6x=-4
4x²-28x+49+x²-6x=-4
5x²-34x+53=0
x=[34±√(1156-1060)]/10
x=[34±√96]/10
X=[34±4√6]/10
x=(17±2√6]/5
X=(17-2√6)/5 or (17+2√6)/5
When x=(17-2√6)/5
Y=2[(17-2√6)/5-7=(34-4√6)/5-7
=(34-4√6-35))/5=-(1+4√6)/5
If x=(17+2√6)/5
What would be y?
I have feeling that the circle x^2− 10x + y^2 + 4x = −4 should have been
x^2− 10x + y^2 + 4y = −4, (why would you have two x terms ?)
then subbing in y=(2x-7)
x^2 - 10x + (2x-7)^2 + 4(2x-7) = -4
x^2 - 10x + 4x^2 - 28x + 49 + 8x - 28 + 4 = 0
5x^2 -30x + 25 = 0
x^2 - 6x + 5 = 0
(x - 1)(x - 5) = 0
x = 1 or x = 5
if x=1, then y = -5
if x=5, then y = 3
To find the coordinates of the points of intersection between the line and the circle, we need to solve the system of equations formed by the given equations.
The equation of the line is 2x - y = 7.
The equation of the circle is x^2 - 10x + y^2 + 4x = -4.
To solve this system, we can use the method of substitution.
Let's solve the line equation for y:
2x - y = 7 ==> y = 2x - 7
Now, substitute this expression for y into the equation of the circle:
x^2 - 10x + (2x-7)^2 + 4x = -4
Expanding and simplifying the equation:
x^2 - 10x + 4x^2 - 28x + 49 + 4x = -4
Combining like terms:
5x^2 - 34x + 45 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, factoring is the most convenient method:
(5x - 9)(x - 5) = 0
Now, set each factor equal to zero and solve for x:
5x - 9 = 0 ==> x = 9/5
x - 5 = 0 ==> x = 5
We have found two possible values for x. Now, substitute these values back into the equation of the line to find the corresponding values of y:
For x = 9/5:
y = 2(9/5) - 7 = 18/5 - 7 = -17/5
So one point of intersection is (9/5, -17/5).
For x = 5:
y = 2(5) - 7 = 10 - 7 = 3
So the other point of intersection is (5, 3).
Therefore, the coordinates of the two points of intersection between the line and the circle are (9/5, -17/5) and (5, 3).