College calculus problem?
If the "hyper ellipse" E is defined by x^4/a + y^4/b=1 where a & b are positive constants.
Find a formula for the derivative as a function of x & y
if the tangent line to E at (x0,y0) has equation cx+dy=1, please find c&d.
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Curve : [ x^( 4/a ) ] + [ y^( 4/b ) ] = 1 ............ (1)
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Differentiating w.r.t. x
(4/a) [ x^(4/a - 1) ] + (4/b) [ y^(4/b - 1) ]·( dy/dx ) = 0
(4/b) [ y^(4/b - 1) ]·( dy/dx ) = - (4/a) [ x^(4/a - 1) ]
dy/dx = - b[ x^(4/a - 1) ] / a[ y^(4/b -1) ] ........................... (2)
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Typing the rest is beyond me.
So now I will give you only the steps :
1. Replacing x by x0, and y by y0, in (2),
... we get Slope of tangent ( m ) at ( x0, y0 ).
2. Then the eq of tgt is ... y- y0 = m( x - x0 ).
... Simplify this eq and put it in the form
... ( ... )x + ( ... )y = 1.
3. Comparing this eq with cx + dy = 1,
... c = first bracket ... and ... d = second bracket ............. Ans.
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Happy To Help !
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