Find the points on the lemniscate where the tangent is horizontal. 8(x2 + y2)2 = 81(x2 − y2)

There are four points where the tangent is horizontal: , , , .

The tangent line is horizontal when slope equals zero and the slope around the lemniscate can be determined by implicit differentiation:

()

(1)

By (1):

Then, the equation of the lemniscate is reduced into this form:

And the value of is:

There are four points where the tangent is horizontal: , , , .

We kindly invite to check this question on lemniscates: link

Step-by-step explanation:

The given equation of lemniscate is

Differentiate with respect to x.

We know that is slope of the tangent and slope of a horizontal line is 0, So, .

Substitute this value in the given equation.

Substitute the value of in the given equation.

On solving we get

Therefore, the required points are .

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step-by-step explanation:

step-by-step explanation: