10. x²+6x-5 > 8x+c
x²-2x-5 > c
(x²-2x+1)-6 > c
(x-1)²-6 > c
(x-1)²-6 is minimum when x=1, i.e. c < -6
∴ Solution set is given by { c: c ∈ R, c <
-6 }
10. x²+6x-5 > 8x+c (alternative, better solution)
x²-2x-(5+c) > 0
For the line to not meet the curve, the solution must not have any
real root(s), i.e. b²-4ac < 0
(-2)²+4(1)(5+c) < 0
4c < -24
c < -6
∴ Solution set is given by { c: c ∈ R, c <
-6 }
11b) x+3y=k-1 ⎯⎯ (1)
y²=2x+5 ⎯⎯ (2)
sub (2) into (1): ½(y²-5)+3y=k-1
y²+6y-(3+2k)=0
For the line to meet the curve, the solution must have real
root(s), i.e. b²-4ac ≥ 0
6²+4(1)(3+2k) ≥ 0
48+8k ≥ 0
k ≥ -6
∴ Solution set is given by { k: k ∈ R, k ≥ -6
}
11c) y² = 8x-x² ⎯⎯ (3)
y = kx+2 ⎯⎯ (4)
sub (3) into (4): (kx+2)² = 8x-x²
k²x²+4kx+4=8x-x²
(k²+1)x²+(4k-8)x+4=0
For the line to meet the curve at 2 points, the solution must have
2 real root(s), i.e. b²-4ac > 0
(4k-8)²-16(k²+1) > 0
64k < 48
k < ¾
∴ Solution set is given by { k: k ∈ R, k <
¾ }
You can read and try to do 11d) and 12. (:
