Draw a diagram to show that there are two tangent lines to the parabola of y = x^2 that pass through the point (0, −9).

Question

Draw a diagram to show that there are two tangent lines to the parabola of y = x² that passes through the point (0, −9).

Find the coordinates of the points where these tangent lines intersect the parabola.

Smaller x-value (x,y) =

Larger x-value (x,y) =

Thank you.

Answer 1

y= x^2 has slope = y' = 2x

a line through the point (0,-9) would have a y intercept of -9

and the equation would be y= mx -9

m= the slope of the parabola and the tangent line

y=x^2 = y = mx-9

x^2 -mx + 9 = 0

x^2- 6x + 9 = 0

m=6

y = 6x -9 is an equation of a tangent line through (0,-9) and (3,9)

from symmetry of the parabola about x=0

another point is (-3,9) where a tangent line also runs through (0,-9)

it's equation has slope = 18/-3 = -6

y= -6x-9

larger x value is 3, smaller is -3