Since all the points are on the same line, the slope between each point will be the same.
slope = \(\frac{\text{change in y}}{\text{change in x}}\)
slope between first and second points = \(\frac{(-5)-(3)}{(3)-(-a+2)}=\frac{-8}{1+a} \)
slope between second and third points = \(\frac{(3)-(2)}{(-a+2)-(2a+3)}=\frac{1}{-3a-1} \)
slope between third and first points = \(\frac{(2)-(-5)}{(2a+3)-(3)}=\frac{7}{2a}\)
Let's pick any two and equate them.
\(\frac{7}{2a}=\frac{-8}{1+a} \) Cross - multiply...
(7)(1+a) = (-8)(2a)
7 + 7a = -16a
7 = -23a
-7/23 = a And here is a graph: https://www.desmos.com/calculator/2pdpfrqz05
If the line's slope is 3 and y-intercept is 1 , the slope-intercept form of the line is:
y = 3x + 1
And the equation for the circle is:
x2 + y2 = 1 We want to find what x is when y = 3x + 1. So substitute 3x + 1 in for y.
x2 + (3x + 1)2 = 1
x2 + (3x +1)(3x + 1) = 1
x2 + 9x2 + 6x + 1 = 1
10x2 + 6x = 0 Factor out an x from both terms.
x(10x + 6) = 0 Set each factor equal to zero and solve for x .
x = 0 or x = -3/5
Now we can plug these x values into the equation of the line to find the y coordinate of the intersection points.
When x = 0 , When x = -3/5 ,
y = 3(0) + 1 y = 3(-3/5) + 1
y = 1 y = -4/5
So...the coordinates of the two points are: (0, 1) and (-3/5, -4/5)
Here's a graph: https://www.desmos.com/calculator/0x0hcauvxk