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Find the equations of the tangent and the normal at the point ‘t’ on the curve x = asin3t, y = bcos3t.

Solution :

We have, x = asin3t, y = bcos3t

  dxdt = 3asin2tcost  and, dydt  = 3bcos2tsint

   dydx = dy/dtdx/dt = ba costsint

So, the equation of the tangent at the point ‘t’ is

y – bcos3t = (dydx)(x – asin3t)

or, y – bcos3t = ba costsint(x – asin3t)

or, bx cos t + ay sin t = ab sin t cos t

The equation of the normal at the point ‘t’ is

y – bcos3t = (1(dydx)(x – asin3t)

or, y – bcos3t = (1(bacostsint)(x – asin3t)

or, ax sin t – by cos t = a^2 sin^4 t – b^2 cos^4 t


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