Prove that sum of intercepts of the tangent at any poin to the curve represented by `x = 3cos^(4)0 " & " y=3sin^(4)0` on the coordinate axis is consta

Question

Prove that sum of intercepts of the tangent at any poin to the curve represented by `x = 3cos^(4)0 " & " y=3sin^(4)0` on the coordinate axis is consta

Prove that sum of intercepts of the tangent at any poin to the curve represented by `x = 3cos^(4)0 " & " y=3sin^(4)0` on the coordinate axis is constant.

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Let P `(3cos^(4)0 . 3sin^(4)0)` be a variable point on the given curve.
`rArr " " m=(dy)/(dx) =((dy)/(d0))/((dx)/(d0)) =(3.4sin^(3) 0.cos0)/(-3.4 cos^(3) 0sin0)=- tan^(2) 0`
`rArr " "" equation of tangent at point P is "`
`y-3sin^(4)0 =- tan^(2)0 (x-3cos^(4)0)`
`rArr " "(x)/(3cos^(2)0) +(y)/(3sin^(4)0) =` ltbegt `rArr` sum of x- axis intercept and y-axis intercept `=3cos^(2)0 +3sin^(2)0 =3` (which is constant)