Determine the set of values of k for which the line 2y = x + k does not intersect the curve y=x2 −4x+7.
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Given:
The equation line 2y = x + k, the curve y=x2 −4x+7
To find:
Determine the set of values of k for which the line 2y = x + k does not intersect the curve y=x2 −4x+7
Solution:
From given, we have,
The equation line 2y = x + k, the curve y = x² − 4x + 7
The minimum of the quadratic occurs when x = -b/(2a) = 4/2 = 2
y = x² − 4x + 7
f(2) = 2² − 4(2) + 7
f(2) = 4 - 8 + 7 = 3
Minimum at (2, 3)
The line equation where 2y = x + k passing through (2, 3) is
2 × 3 = 2 + k
k = 4
Equation:
2y = x+4
y = x/2 + 2
Therefore, the value of k should be k < 4
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