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Find the set of values of k for which the line y=kx−4 intersects the curve y=x at two distinct points.

A) k<−1 or k>1
B) k<1 or k>−1
C) k<4 or k>−4
D) k<−4 or k>4

1 Answer

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Final answer:

The set of values of k for which the line y = kx - 4 intersects the curve y = x at two distinct points is when k < -1 or k > 1. This excludes the possibility of k being 1 or -1, which would result in parallel lines that never intersect the curve.

Step-by-step explanation:

To determine the set of values of k for which the line y = kx - 4 intersects the curve y = x at two distinct points, we will set both equations equal to each other because at the point of intersection, the y-values of the line and the curve are the same.

y = kx - 4
y = x

Setting the two y-values equal gives us:

kx - 4 = x
kx - x = 4
x(k - 1) = 4

For the line to intersect the curve at two distinct points, the value inside the parentheses (k - 1) must not be zero, since x can be any value other than zero for a non-horizontal line. Therefore, k - 1 ≠ 0, which implies that k ≠ 1. If k were 1, the line would be y = x - 4, which is a parallel line to y = x and would never intersect it. Similarly, if k were -1, the lines would also be parallel, this time as mirror images of each other with respect to the x-axis. Hence, the values for k for which the line and the curve intersect at two distinct points are when k < -1 or k > 1, which corresponds to option A.

User Sukhjeevan
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