F4 MATHs

1.The line x+y =k touches the parabola y=x^2 -7x+14

a)Find the value of k

x+y =k----->y=k-x----------1
y=x^2 -7x+14----------------2
Sub 1 into 2
k-x=x^2-7x+14
x^2-6x+(14-k)=0
The line x+y =k touches the parabola y=x^2 -7x+14 and hence the intersection between two equations is only one
delta=0
6^2-4(14-k)=0
4k=20
k=5


b)Find the coordinates coordinates of the point of contact
put k=5 into x^2-6x+(14-k)=0
x^2-6x+9=0
(x-3)^2=0
x=3 and y=5-3=2
Therefore, the coordinates coordinates of the point of contact is (3,2)

2.The line y=kx-2 touches the parabola y=x^2-4x+7.

a)Find the possible values of k

y=kx-2---------1
y=x^2-4x+7-----------2
Sub 1 into 2
kx-2=x^2-4x+7
x^2-(4+k)x+9=0
The line y=kx-2 touches the parabola y=x^2-4x+7 and hence the intersection between two equations is only one
delta=0
(4+k)^2-4(9)=0
16+8k+k^2-36=0
k^2+8k-20=0
(k+10)(k-2)=0
k=-10 or k=2
b)If k takes the negative value, find the coordinates of the point of contact.
If k takes the negative value,then k=-10
put k=-10 into x^2-(4+k)x+9=0
x^2+6x+9=0
(x+3)^2=0
x=-3 and y=(-10)(-3)-2=28
Therefore,the coordinates of the point of contact=(-3,28)