Question

20. Use Barrow's a, e method to determine the slope of the tangent line to the curve x^3 + y^3=C^3

Answer 1

Answer: Slope would be,

`-(x^2)/(y^2)`

Explanation:

Here, the given curve,

`x^3 + y^3=C^3`

`\implies x^3 + y^3 - C^3=0`

In Barrow's method,

Steps are as follows,

Step 1 : put, x = x - e, y = y - a

`(x-e)^3 + (y-a)^3 - C^3=0`

`x^3-3x^2e+3xe^2-e^3+y^3-3y^2a+3ya^2-a^3+C^3=0`

Step 2 : Reject terms which do not contain a or e,

`-3x^2e+3xe^2-e^3-3y^2a+3ya^2-a^3=0`

Step 3 : Reject all terms in which a or e have exponent greater than 1,

`-3x^2e-3y^2a=0`

Step 4 : Find the ratio of a : e,

`-3y^2a=3x^2e`

`\implies (a)/(e)=-(x^2)/(y^2)`

Hence, the slope of the given curve is
`-(x^2)/(y^2)`