Posted by **albian** on Friday, August 28, 2009 at 8:40pm.

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 ?

- maths -
**bobpursley**, Friday, August 28, 2009 at 9:43pm
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=....

- maths -
**Reiny**, Friday, August 28, 2009 at 9:50pm
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!)

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