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?
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y = 2(x-3)(x-5) = 2x^2 -16x + 30
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The zeroes are 3 & 5, and the vertex is (4,-2)
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Find the equation of the "similar" parabola with vertex (0,-2)
Its zeroes are -1 & +1.
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y = ax^2 + bx + c
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At (-1,0): 0 = a - b + c
At (1,0): 0 = a + b + c
----------------------------- Subtract
b = 0
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At (0,-2): -2 = c
--> y = 2x^2 - 2
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Find y where x = -3 or +3
y = 2*9 - 2 = 16
---> k = 16
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Might be a "better" or shorter way, but k = 16 regardless.
Answer by ikleyn(52879)  (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(13209)  (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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