The graph of the line y = 3x + a intersects the graph of the parabola y = x^2 + x in two points. If the distance between these points is sqrt(30), what is the value of a? Express your answer as a common fraction.
+128474 Set the equations = to find the x coordinates (in terms of a) of the intersections of these two functions
3x + a = x^2 + x rearrange as
x^2 - 2x = a complete the square on x
x^2 -2x + 1 = a + 1
(x - 1)^2 = a + 1 take both roots
x - 1 = ± √ (a + 1)
x = ± √ (a + 1) + 1
Plugging these two x values into the equation of the line we have that
y = 3 ( sqrt (a + 1) + 1) + a or y = 3 ( -sqrt (a + 1) + 1) + a
y = 3sqrt (a + 1) + a + 3 y = -3sqrt (a + 1) + a + 3
Using the square of the distance formula we have that
( (sqrt (a + 1) + 1) - ( -sqrt (a + 1) + 1) )^2 + ( (3sqrt (a + 1) + a + 3) - (-3sqrt (a + 1) + a + 3))^2 = 30
Simplify
( 2 sqrt ( a + 1) )^2 + ( 6 sqrt (a + 1 )^2 = 30
4 (a + 1) + 36 (a + 1) = 30
40a + 40 = 30
40a = -10
a = -1/4
Here's a graph :
Points A and B are the intersections of the functions.......the midpoint of the segment connecting A and B = (1, 2.75)
A circle with this center and a diameter of sqrt (30) = radius of sqrt (30) /2 will pass through both points
