Q.201. Normal to a parabola.

Question 201.
The normal of the graph of y= (x-2)^2 at the point A(-3,1) meet the graph again at B. Find the angle at B between the graph and the chord AB.

Answer 201.
y = (x + 2)^2
=> dy/dx = 2(x + 2)
=> (dy/dx) at (-3, 1) = -2
=> slope of normal = 1/2
=> eqn. of normal through (-3, 1) having slope 1/2 is
y - 1 = (1/2)(x + 3)
=> y = x/2 + 5/2

Solving this with the eqn of parabola
(x + 2)^2 = x/2 + 5/2
=> 2x^2 + 7x + 3 = 0
=> (2x + 1)(x + 3) = 0
=> B is (-1/2, 9/4)

Slope of AB = (9/4 -1)/(-1/2 +3) = 1/2 = m
Slope of graph at B(-1/2, 9/4) = 2(-1/2 + 2) = 3 = m'
Reqd. angle
= arctan l(m - m') / (1 + mm')l
= arctan l(1/2 - 3) / (1 + 3/2)l
=arctan 1
= π/4.

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