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Q2- angle between the tangents drawn from point (1,4) to the parabola y^2=4ax ??

please eleborate and also give the formulas if used

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y = mx + a/m is the tangent to the parabola y^{2} = 4ax

Let us take value of 'a' to be 1 here (such that answer is shown cleaner. Won't make any difference)

For y^{2} = 4x, a = 1

=> y = mx + 1/m is the tangent

If it passes through (1, 4),

4 = m + 1/m

=> m^{2} - 4m + 1 = 0

The roots m1 and m2 are the slopes of the tangents.

Use sum of roots and product of roots in a Quadratic Equation.

=> m1 + m2 = 4 and m1m2 = 1

Let us calculate value of m1 - m2

=> m1 - m2

= √[(m1 + m2)^{2} - 4m1m2]

= √(16 - 4)

= 2√3

If θ = angle between the tangents, then

tanθ

= l (m1 - m2) / (1 + m1m2) l

= l 2√3 / (1 + 1) l

= √3

=> θ = tan^{-1}√3 = π/3.

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