Question

consider the line y=6x-k and the parabola y=x^2

i) for what value of k is the line y=6x-k a tangent to the parabola y=x^2 ?

ii) the line y=6x-k intersects the parabola in two distinct places. what is the largest integer value that k can take ?

Answer 1

i. at x=3, the slope of the parabola is 6, so at 3, the parabola has a y value of 9, which means if the line y=6x+k is =9, then k=....

Answer 2

i)
using Calculus, this is easy
if y = x^2
dy/dx = 2x
but the slope of the line is 6
so 2x = 6
x = 3
if x=3 then y=9 from the parabola
so the point (3,9) must also be on the line
9 = 6(3)-k
k = 9

ii) let's intersect
y = x^2 and y = 6x-k
then x^2 - 6x + k = 0

to have 2 distinct roots, the discriminat must be positive
so b^2 - 4ac > 0
36 - 4(k) > 0
k < 9

since k=9 produces the tangent, and you asked for the largest integer that k could have,
k = 8
(strange question!)