First slide
Tangents and normals
Question

A curve is given by the equations x=sec2θ,y=cotθ. If the tangent at P where θ=π4 meets the curve again at Q, then [PQ] is, where [ . ] represents the greatest integer function _____

Moderate
Solution

dydx=-12cot3θ=12atθ=π4

Also, the point P for θ=π4is2,1

Equation of tangent is 

y-1=-12x-2orx+2y-4=0       (1)

This meets the curve whose Cartesian equation on eliminating 

θbysec2θ-tan2θ=1 is

y2=1x-1                                                (2)

Solving (1) and (2) we get y=1,-12

x=2,5

Hence, P is (2, 1) as given and Q is 5,-12. Therefore,

PQ=454=352

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