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= x^2 -6x + 1

y' = 2x -6

a line through (0,0) has an equation y=mx where m is the slope = tangent line slope

slope of the tangent line is 2x-6 for the parabola

the points of tangent lines to the parabola are (1,-4) and (-1, 8)

the tangent lines' equations are y=-4x and y=-8x

for (1,-4) the slope of the line through the origin = (-4-0)/(1-0) = -4

at the point (1,-4) the parabola's slope is y' =2x-6 = 2(1)-6 = -4

for (-1,8) the slope of the line through the origin = (8-0)/(-1-0) = -8

at the point (-1,8) the parabola's slope is y' = 2x -6 = 2(-1) -6 = -2-6=-8

it might help to graph the parabola and the points

the vertex of the parabola is (3,-8)

y =x^2 -6x +1

y' =2x -6 = 0

2x =6

x=3

y=(3)^2 -6(3) +1 = 9-18+1 =-8