Question

Cxis and CyCy be the chords of curvatureParallel to the axis at any point curve y=a.e^x÷y ​

Answer 1

We cannot determine the specific chords of curvature parallel to the axis for the given curve y = a * $e^{\frac{x}{y} }$.

Explanation:

To analyze the given curve, let's start by finding the equation of the curve and then determining the chords of curvature.

The equation of the curve is y = a * $e^{\frac{x}{y} }$. Here, 'a' is a constant and e represents the mathematical constant approximately equal to 2.71828.

To find the chords of curvature parallel to the axis, we need to determine the points on the curve where the curvature is constant. The curvature of a curve at a point is given by the formula: K = $\frac{|y^{''}| }{(1+(y^{'} )^{2}) ^{\frac{3}{2} } }$, where y' represents the first derivative of y with respect to x and y'' represents the second derivative of y with respect to x.

Differentiating y = a*$e^{\frac{x}{y} }$ with respect to x, we get:

y' = $\frac{(a*e^{\frac{x}{y} }) *(1-\frac{x}{y^{2} }) }{y}$

Differentiating y' with respect to x, we get:

y'' = $\frac{{(a*e^{\frac{x}{y} }) *(y^{2}-2xy) }}{y^{4} }$

Now, to find the chords of curvature parallel to the axis, we need to equate y'' to zero. However, in this case, y'' cannot be equated to zero as it involves the term $y^{4}$ in the denominator.

Therefore, we cannot determine the specific chords of curvature parallel to the axis for the given curve y = a * $e^{\frac{x}{y} }$.

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