Find an equation of the tangent line at each given point on the curve. x = 2 ? 3

Question

Find an equation of the tangent line at each given point on the curve. x = 2 ? 3 cos(?) y = 3 + 2 sin(?) 1) (2,5) 2) ((4+3sqrt(3))/2,2) 3) (-1,3) The tangent line equation for 3) starts with x= instead of y= like the first two points. x = 2 ? 3 cos(?) y = 3 + 2 sin(?) 1) (2,5) 2) ((4+3sqrt(3))/2,2) 3) (-1,3) The tangent line equation for 3) starts with x= instead of y= like the first two points. x = 2 ? 3 cos(?) y = 3 + 2 sin(?) 1) (2,5) 2) ((4+3sqrt(3))/2,2) 3) (-1,3) The tangent line equation for 3) starts with x= instead of y= like the first two points.

Explanation / Answer

i am taking theta= t

x= 2-3cost

dx/dt = 3sint

y= 3+2sint

=>dy/dt = 2cost

dy/dx= (dy/dt)/(dx/dt) = 2cost/ 3sint = (2/3)cot t

i) at (2,5) , t= pi/2

=> dy/dx= 0

=> equation of tangent is

y-5 = 0(x-2)

=> y=5

ii)at ((4+3sqrt(3))/2,2)

t= -pi/6

=> slope = dy/dx= -2/sqrt3

=> equation of tangent is

y- 2 = (-2/sqrt3)[x- 4-3sqrt(3))/2]

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