Find a set of values of #k# for which the line meets the curve at two distinct points?

Find the set of values of #k# for which the line #y = 2x -k# meets the curve #y = x^2 + kx - 2# at two distinct points.

1 Answer
Jim G.
Sep 13, 2017

#k<2" or "k>6#

Explanation:

#"equating the line and the parabola"#

#rArrx^2+kx-2=2x-k#

#"rearrange and equate to zero"#

#x^2+kx-2x-2+k=0#

#rArrx^2+x(k-2)+(k-2)=0#

#"with "a=1,b=(k-2),c=(k-2)#

#"for the equation to have 2 real distinct roots"#

#"the "color(blue)"discriminant "Delta>0#

#Delta=b^2-4ac=(k-2)^2-4(k-2)#

#color(white)(xxxxxxxxx)=k^2-4k+4-4k+8#

#color(white)(xxxxxxxxx)=k^2-8k+12#

#"to solve "k^2-8k+12>0" sketch the graph"#

#"solve "k^2-8k+12=0#

#rArr(k-6)(k-2)=0rArrk=6,k=2#

#"coefficient of "k^2>0rArr" minimum " uuu#

#"consider the parts of the graph above the k-axis"#
graph{x^2-8x+12 [-10, 10, -5, 5]}

#"solution is "k<2" or "k>6#

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