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Draw a diagram to show that there are two tangent lines to the parabola
y = x2
that pass through the point
(0, −25).
Find the coordinates of the points where these tangent lines intersect the parabola.

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  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).

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