Draw a diagram to show that there are two tangent lines to the parabola

Question

y = x2
that pass through the point
(0, −25).
Find the coordinates of the points where these tangent lines intersect the parabola.

Answer 1

the parabola y=x² has it's vertex at (0,0).
Tangent to parabola has slope
dy/dx=2x, at point (±x, x²).

For tangents to pass through (0,-25), we have the slope joining (0,-25) and (x,x²) equal to 2x.

Slope=(y2-y1)/(x2-x1)
=>
(x^2-(-25))/(x-0)=dy/dx=2x
solving for x, we get
x=+5 or x=-5.
Thus the tangent points are
(5,25) and (-5,25).