Let (x0, y0) be a point on the parabola x^2 = 4py.

Show that the equation of the line tangent to the parabola at (x0, y0) is y = ((x0)/(2p))x - y0

Let (x0, y0) be a point on the parabola x^2 = 4py.

Show that the equation of the line tangent to the parabola at (x0, y0) is y = ((x0)/(2p))x - y0

Differentiate to find the tangent: 2x=4pdy/dx so the tangent dy/dx=x/2p.

At (x₀,y₀), dy/dx=x₀/2p. This is the slope of the tangent line.

The equation of the tangent is y-y₀=(x₀/2p)(x-x₀), so **y=(x₀/2p)(x-x₀)+y₀**.

Note that the expected equation as given in the question cannot be correct because it doesn’t pass through the tangent point (x₀,y₀).

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