What are the two tangent lines to the parabola y=x^2-6x+1 through the point (0,0).

Question

What are the two tangent lines to the parabola y=x^2-6x+1 through the point (0,0).

Answer 1

y' = 2x-6 = the derivative of y(x) = the slope of the tangent line

the tangent line through the origin is either y=-4x or y=-8x

there are two lines through the origin that are tangent to the parabola

y=-4x is tangent at the point (1,-4), y=-8x is tangent at the point (-1,8)

y' =2x-6 = slope of the tangent line = y/x

y/x = 2x-6

8/1 = 2(-1)-6

8 = 8

or -4/1= 2(1)-6

-4 =-4

sketch the parabola and you can see 2 lines could go through the origin with tangency at points in quadrant II and IV. You can make an educated estimate close to the above