SOLUTION: The horizontal line y = k intersects the parabola with equation y = 2(x-3)(x-5) at points A and B. If the length of line segment AB is 6, what is th value of k?

Question 1146943: The horizontal line y = k intersects the parabola with equation y = 2(x-3)(x-5) at points A and B. If the length of line segment AB is 6, what is th value of k?
Found 3 solutions by Alan3354, ikleyn, greenestamps:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
The horizontal line y = k intersects the parabola with equation y = 2(x-3)(x-5) at points A and B. If the length of line segment AB is 6, what is th value of k?
------------------
y = 2(x-3)(x-5) = 2x^2 -16x + 30
---
The zeroes are 3 & 5, and the vertex is (4,-2)
--
Find the equation of the "similar" parabola with vertex (0,-2)
Its zeroes are -1 & +1.
----
y = ax^2 + bx + c
--
At (-1,0): 0 = a - b + c
At (1,0): 0 = a + b + c
----------------------------- Subtract
b = 0
===========
At (0,-2): -2 = c
--> y = 2x^2 - 2
-------------------
Find y where x = -3 or +3
y = 2*9 - 2 = 16
---> k = 16
================================
Might be a "better" or shorter way, but k = 16 regardless.

Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.

The equation 2*(x-3)*(x-5) = k is equivalent to 2x^2 - 16x + 30 = k, or 2x^2 - 16x + (30-k) = 0, or x^2 - 8x + = 0. According to Vieta's theorem, the sum of the roots is equal to coefficient at "x" with the opposite sign, i.e. 8. The difference of the roots is equal to 6 (given). If "a" is the greater root, then the other root is (8-a), and the difference between them is 6 = a - (8-a) = 2a - 8, which implies 2a = 6 + 8 = 14, a = 7. Thus the greater root is a= 7, while the smaller root is 8-a = 8-7 = 1. The product of the roots, 7*1, is equal to the constant term , according to Vieta's theorem (again (!) ). Thus you have this equation for k = 7, which implies 30 - k = 14. Hence, k = 30-14 = 16.

Solved.

VERY GOOD problem on Vieta's theorem (!)

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

The parabola has zeros at x=3 and x=5, so the axis of symmetry is x=4.

For the length of AB to be 6, the x values of points A and B have to be 3 either side of the axis of symmetry; so the x values of points A and B are 1 and 7.

Substitute either x=1 or x=7 into the equation to find the y value at those points, which is the value of k the question asks for.

ANSWER: k = 16
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