Line Equations : Tangents & Normals
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I) Find the equation of the tangent to y=x(1-x) at x=1
ii) Find the equation of the normal to y=x(1-x) at x=1
iii) Find the equations of the tangents to y=x(1-x) that pass through (-1, 1/4)
At x =1: y = 1*(1-1) = 0
Hence, given point (x1, y1)= (1,0)
dy/dx = y' = 1 - 2x
y'|(at x = 1) = 1 - 2*1 = -1
Hence, Eqn of tangent:
y -y1 = y'(x-x1)
=> y - 0 = (-1)*(x-1)
=> x + y = 1 --Answer
Eqn of normal:
(x-x1) + y'(y-y1) = 0
=> (x-1) + (-1)*(y-0) = 0
=> x-1 -y = 0
=> x - y = 1 ...
Tangents and normals are calculated.