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 3\(\sqrt{30}\), what is the value of a? Express your answer as a common fraction.

**Please include the solution instead of just the answer! That would be much appreciated. Also, if any moderators would help, that would be great**

Imcool Feb 1, 2023

#1**0 **

To find the value of a, we need to determine the two points of intersection between the two functions. Let's call the two points of intersection (x1,y1) and (x2,y2).

Setting y = 3x + a equal to y = x^2 + x, we get:

3x + a = x^2 + x

3x + a - x^2 - x = 0

x^2 + 2x + (3-a) = 0

We can now use the quadratic formula to find the solutions for x:

x = (-b ± √(b^2 - 4ac)) / 2a

x = (-2 ± √(2^2 - 4 * 1 * (3-a))) / 2 * 1

x = (-2 ± √(4 + 4a - 4 * 3)) / 2

x = (-2 ± √(4a - 8)) / 2

Since the two points of intersection are distinct, the square root must be real and nonnegative, so:

4a - 8 ≥ 0

a ≥ 2

The two points of intersection can now be found by substituting the values of x back into y = 3x + a:

y1 = 3x1 + a = 3 * (-2 + √(4a - 8)) / 2 + a = -3 + 3√(4a - 8) + a

y2 = 3x2 + a = 3 * (-2 - √(4a - 8)) / 2 + a = -3 - 3√(4a - 8) + a

Finally, the distance between the two points is given as 3 * sqrt(15), so:

√((x1 - x2)^2 + (y1 - y2)^2) = 3 * √(30)

√((-2 + 2√(4a - 8))^2 + (3 + 3√(4a - 8) - (-3 - 3√(4a - 8)))^2) = 3 * √(30)

√(4 + 6√(4a - 8)) = 3 * √(30)

4 + 6√(4a - 8) = 15

6√(4a - 8) = 11

√(4a - 8) = 11 / 6

√(4a - 8) = √(11 / 6)

4a - 8 = 11 / 6 * 11 / 6

4a = 11 / 6 * 11 / 6 + 8

4a = 121 / 36 + 144 / 36

4a = 265 / 36

a = 265 / 36 * 9 / 4

a = 765 / 144

So the value of a is 765/144. This is already in the form of a common fraction, so this is our final answer.

Guest Feb 1, 2023