Normal and coordinate ranges of a logarithmic curve
Normal and coordinate ranges of a logarithmic curve
Junior College 2
7 marks
A curve is defined parametrically by \(x=\ln(t+1)\) and \(y=\ln(5-t)\), where \(t\) takes all real values for which both expressions are defined.
Using calculus, find the gradient of the normal to the curve at the point where \(t=1\).[3]
Eliminate \(t\) to obtain a Cartesian equation of the curve. State the exact ranges of \(x\) and \(y\), justifying whether their boundary values are included.[4]
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Answer:(a) Normal gradient \(=2\) (b) \(e^x+e^y=6\), \(x<\ln6\), \(y<\ln6\)