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let C be the curve y=(k+2)x2 -2kx +1, where k is a real number and k≠-2.
(a) find the range of values of k such that C does not intersect with the x-axis.
If C does not intersect x-axis.
That means determinant △ of (k+2)x2 - 2kx + 1 = 0 < 0
discriminant △ < 0
(-2k)2 -4(k+2)(1) < 0
4k2-4k-8 < 0
(k+1)(k-2) < 0
-1 < k < 2
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(b)(i) if C cuts the x-axis at two points P and Q and PQ =2, find the value(s) of k.
Let P be (p, 0) and Q be (q, 0) So the roots be p and q.
Sum of roots = -(-2k)/(k+2)
p + q = 2k/(k+2) ... (1)
Product of roots = 1/(k+2)
pq = 1/(k+2) ... (2)
Given the length of PQ = 2
|p-q| = 2
(p-q)2 = 4
p2 - 2pq + q2 = 4
(p2 + 2pq + q2) - 4pq = 4
(p+q)2 - 4(pq) = 4
[2k/(k+2)]2 - 4[1/(k+2)] = 4 【from (1), (2)】
4k2/(k+2)2 - 4/(k+2) = 4
4k2 - 4(k+2) = 4(k+2)2
4k2 - 4k - 8 = 4k2 + 16k + 16
20k + 24 = 0
k = -6/5
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(ii) hence, find the coordinates of the mid-point of PQ
x-coordinate the mid-point of PQ
= (p+q)/2
= [2k/(k+2)]/2
= k/(k+2)
= (-6/5) / (-6/5 + 2)
= (-6/5)(5/4)
= -3/2
y-coordinate of mid-point of PQ
= (0+0)/2
= 0
So the midpoint of PQ is (-3/2, 0)

bi)
PQ = 2
implies difference of roots = 2
i.e. a-b = 2 (for a&gt;b)

also, from the equation, we get
a+b = 2k/(k+2), and ab = 1/(k+2)

so, we can do it like this:

a+b = 2k/(k+2)

(a+b)^2 = [ 2k/(k+2) ]^2

a^2 + 2ab + b^2 = [ 2k/(k+2) ]^2

a^2 - 2ab + b^2 +4ab = [ 2k/(k+2) ]^2

(a-b)^2 + 4ab = [ 2k/(k+2) ]^2

4 + 4/(k+2) = [ 2k/(k+2) ]^2

4(k+2)^2 + 4(k+2) - 4k^2 = 0

4k^2 + 16k + 16 + 4k + 8 - 4k^2 =0

20k + 24 = 0

k = -6/5

ii) Subs. k=-6/5 into the result we got before

a+b = 2k/(k+2) = -3 , and
a-b = 2

we get a = -1/2, b = -5/2

then, x-coordinate of mid-pt of PQ = (-1/2 + -5/2)/2 = -3/2

so, mid-point of PQ = (-3/2,0)

你已計了(a)

(b) let the root be A and B
consider the absolute of the roots, that is abs[A-B]=2
(A-B)^2 = 4
A^2 + B^2 -2A*B = 4
(A+B)^2 - 4*A*B = 4
sub of roots = A+B = 2k/(k+2)
product of roots = A*B = 1/(k+2)
then, 4*k^2/(k+2)^2 -4/(k+2) = 4
k^2 - (k+2) = (k+2)^2
k = -6/5

mid-point of PQ = (A+B)/2 = 2k/(k+2)/2=2*(-6/5)/(-6/5+2)/2=-1.5

therefore mid-point of PQ = (-1.5,0)