Let y= mx-2 be the tangent line to y=x^2-4x+2 at x=a.
Then slope, [tex]m=\frac{dy}{dx} at x=a = 2a-4.[/tex]
Hence the equation is y=(2a-4)x-2
Let's find y-coordinate at x=a using tangent line and curve.
Using tangent line y at x=a is (2a-4) a -2 [tex]=2a^{2}-4a-2[/tex]
Using given curve y-coordinate at x=a is [tex]a^{2}-4a+2[/tex]
Let's equate these 2 y-coordinates,
[tex]2a^{2} -4a-2 = a^{2} -4a+2[/tex]
[tex]2a^{2}-a^{2} = 2+2[/tex]
[tex]a^{2}=4[/tex]
a=2 or -2.
If a=2, [tex]m=2a-4 = 2*2-4=0[/tex]
If a=-2,[tex]m= 2(-2)-4 = -8[/tex]
Hence m values are 0 and -8.
