The equation of the lemniscate is $\"\"$ .

Slope of the tangent is derivative of the function.

Find the derivative of the curve .

Consider $\"\"$ .

Differentiate on each side with respect to $\"\"$ . \ \

$\"\"$ .

Power rule of derivatives : $\"\"$ .

$\"\"$

$\"\"$ .

The curve has horizontal tangent when the slope of the tangent is zero.

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Substitute $\"\"$ in the lemniscate equation $\"\"$ .

$\"\"$

Solve the two equations $\"\"$ and $\"\"$ .

Subtract the two equations.

$\"\"$

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Therefore points on the curve where the tangent is horizontal are $\"\"$ .

Points on the lemniscate where the tangent is horizontal are $\"\"$ .