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The graphs y =2x2 and y = 4c2 - 2x2, where c is a positive constant, h
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23 Feb 2022, 08:15
23 Feb 2022, 08:15
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The graphs \(y =2x^2\) and \(y = 4c^2 - 2x^2\), where c is a positive constant, have two intersecting points. If a line segment connecting the two points has a length of 2, what is the value of c?
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Re: The graphs y =2x2 and y = 4c2 - 2x2, where c is a positive constant, h
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23 Feb 2022, 23:52
23 Feb 2022, 23:52
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Carcass wrote:
The graphs \(y =2x^2\) and \(y = 4c^2 - 2x^2\), where c is a positive constant, have two intersecting points. If a line segment connecting the two points has a length of 2, what is the value of c?
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Great question!
\(y =2x^2\) is a parabola (opening upwards, because a > 0) with its vertex at origin (0, 0)
Now, Since c > 0
\(y = - 2x^2 + 4c^2\) will be a parabola (opening downwards, because a < 0) with its vertex at y-axis and \(4c^2\) units above the x-axis i.e. (0, \(4c^2\))
In order to find out the point of intersection, we can equate the y-values;
\(2x^2 = - 2x^2 + 4c^2\)
\(4x^2 = 4c^2\)
\(x^2 = c^2\)
\(x = c\)
We also know that the parabola is a symmetric figure, which means if the line segment connecting the two points of intersection has a length of 2 then the length will be equally divided (refer to the figure below).
Hence, \(c = 1\)
Attachment:
The graphs y =2x^2.jpg [ 1.04 MiB | Viewed 1917 times ]
gmatclubot
Re: The graphs y =2x2 and y = 4c2 - 2x2, where c is a positive constant, h [#permalink]
23 Feb 2022, 23:52
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