5. The curve C' is defined by x = cos t, y = sin 2t, t E [0, 27] (a) Show that C has two tangent lines at the point (0, 0) and find the equations of the tangent lines. (b) Find the points (x, y) at which C has (1) a horisontal and (2) a vertical tangent line. (c) Sketch C.

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A: Corresponding to θ=0, the x and y coordinates will be,

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A: Given :       x=2-πcost, y=2t-πsint

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A: The slope of the tangent is dydx.

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Q: 27. Show that the tangent at any point 0 on the curve x = csec 0, y = c tan is y sin 0=x-c cos 0.

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A: Using method to find locus we get

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A: The given curve and the point is x=sin(t), y=cos2(t) ; 32,14

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A: Solving

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A: Given

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A: The objective is to find the slope of tangent line to the curve at the point (7π, 3π/2).

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Q: 4. Find the equation of the tangent line to y sin(2r) = r cos(2y) at the point (, ).

A: topic - equation of tangent line to curve

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A: Given curve is:  2y3-xy2+cos(xy) = 3 For slope of the tangent, differentiate the curve equation

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A: Given, curve and point

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A: Formula: sin2 t+ cos2 t=1 The given parametric equation is: x(t)=3+ sin(t) and y(t)=-2+ cos(t) where…

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Q: Find the point of intersection of the tangent line to y=sin( 2x) at y=0 and the line r+2y 5. O (5,0)…

A: let us first ind the equation of tangent  line

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A: on solving this we get