The parametric equations of a curve are ( x=acos^3t , y=asin^3t ) where a is a positive constant and 0
(1) show that the equation of the tangent to the curve at the point with parameter t is xsint+ycost=asintcost.
(2) hence show that if this tangent meets the x axis at X and the y axis at Y, then the length of XY is always equal to a.
Answers
Given : The parametric equations of a curve are ( x=acos^3t , y=asin^3t ) where a is a positive constant
To Find : show that the equation of the tangent to the curve at the point with parameter t is xsint+ycost=asintcost.
Solution:
Tangent dy/dx
= (dy/dt)/(dx/dt)
x=acos³t
=> dx/dt = 3acos²t(-sint)
y=asin³t
=> dy/dt = 3asin²t(cost)
Tangent dy/dx = 3asin²t(cost) / 3acos²t(-sint)
= -sint/cost
y - asin³t = ( -sint/cost)(x - acos³t)
=> ycost - asin³tcost = -xsint + acos³tsint
=> xsint + ycost = asin³tcost + acos³tsint
=> xsint + ycost = asin tcost(sin²t + cos²t)
=> xsint + ycost = asin tcost(1)
=> xsint + ycost = asin tcost
QED
Hence Proved
at x axis y = 0
=> xsint + 0 = asin tcost => X = acost
at y axis x = 0
=> 0 + ycost = asin tcost => Y = asin t
length of XY = √(acost)² + (asint)² = a√cos²t + sin²t
= a√1
= a
length of XY = a
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Answer:
xSint + yCost = aSintCost
Step-by-step explanation:
explanation is given in the pic attached
