Question

Consider the curve with equation given by x⁴-x²y+y⁴=24

1. verify that the point (-2,2) lies on the curve.

2. find the equation of the line tangent to the curve at point (-2,2)

3. provide the graphs of the given curve and the tangent line together on the same xy axis.

Answer 1

We are given equation as

(1)

we can plug x=-2 and y=2 in left side

and check whether it is same as right side

Since, left side= right side

so,

the point (-2,2) lies on the curve

(2)

we can find derivative with respect to x

now, we can plug x=-2 and y=2

now, we can solve for y'

now, we can use point slope form of line

now, we can solve for y

............Answer

(3)

we can draw graph