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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

   dy\over dx = dy/dt\over dx/dt = -b\over a cos t\over sin t

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

y – b cos^3 t = (dy\over dx)(x – a sin^3 t)

or, y – b cos^3 t = -b\over a cos t\over sin t(x – a sin^3 t)

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

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

y – b cos^3 t = (-1\over ({dy\over dx})(x – a sin^3 t)

or, y – b cos^3 t = (-1\over ({-b\over a}{cos t\over sin t})(x – a sin^3 t)

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


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