Tangent line equation.Given x = 5cost, y =3sint, what is the equation of the tangent line if t=pi/4.
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It is given that x = 5*cos t and y = 3*sin t. We have to find the equation of the tangent if t = pi/4
When t = pi/4 , x = 5*(1/sqrt 2) and y = 3/sqrt 2
dx/dt = -5*sin t and dy/dt = 3*cos t
dy/dx = (dy/dt)/(dx/dt)
=> 3*cos t/ -5*sin t
=> (-3/5)/tan t
at t = pi/4
=> -3/5
The equation of the tangent is (y - 3/sqrt 2)/(x - 5/sqrt 2) = -3/5
=> 5y - 15/sqrt 2 = -3x + 15/sqrt 2
=> 3x + 5y - 30/sqrt 2 = 0
The equation of the tangent is 3x + 5y - 30/sqrt 2 = 0
We'll use the chain rule to differentiate:
dy/dx = (dy/dt)/(dx/dt)
dx/dt = (d/dt)(5cost)
dx/dt = -5 sin t
dy/dt = (d/dt)(3sint)
dy/dt = 3 cos t
dy/dx = 3 cos t/-5 sin t
dy/dx = 3 cos(pi/4)/-5 sin (pi/4)
dy/dx = 3/-5
At t = pi/4, we'll get the point (x(t),y(t))= (5cos(pi/4),3sin(pi/4))
(x(t),y(t))= (5sqrt2/2,3sqrt2/2)
The equation of the tangent line is:
y - 3sqrt2/2 = m(x - 5sqrt2/2)
y - 3sqrt2/2 = (-3/5)(x - 5sqrt2/2)
y = (-3/5)(x - 5sqrt2/2) + 3sqrt2/2
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