A line has equation \(y = 2x + 3\) and a curve has equation \(y = cx^2 + 3x - c\), where \(c\) is a constant.
Showing all necessary working, determine which of the following statements is correct.
A. The line and curve intersect only for a particular set of values of \(c\).
B. The line and curve intersect for all values of \(c\).
C. The line and curve do not intersect for any values of \(c\).
Solution
To find the points of intersection, set the equations equal: \(2x + 3 = cx^2 + 3x - c\).
Rearrange to form a quadratic equation: \(cx^2 + (3 - 2c)x - (c + 3) = 0\).
The discriminant of a quadratic \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\).
Here, \(a = c\), \(b = 3 - 2c\), and \(c = -(c + 3)\).
Calculate the discriminant: \((3 - 2c)^2 - 4c(-c - 3)\).
Simplify: \((3 - 2c)^2 + 4c(c + 3) = 9 - 12c + 4c^2 + 4c^2 + 12c = 8c^2 + 9\).
The discriminant \(8c^2 + 9\) is always positive for all values of \(c\), indicating two real roots.
Thus, the line and curve intersect for all values of \(c\).
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