Answer
$(-2.29129,1.24514);\hspace{1cm}(-2.29129,-1.24514)$
$(2.29129,-1.24514);\hspace{1cm}(2.29129,1.24514)$
Work Step by Step
The lemniscate equation is:
$2(x^2+y^2)^2=25(x^2-y^2)$
$2x^4+4x^2y^2+2y^4=25x^2-25y^2$
Take the derivative
$8x^3+8xy^2+8x^2y\frac{dy}{dx}+8y^3\frac{dy}{dx}=50x-50y\frac{dy}{dx}$
$\frac{dy}{dx}=\frac{50x-8xy^2-8x^3}{8y^3+8xy^2+50y}$
$50x-8xy^2-8x^3=0$
$x=-2.29129;\hspace{5mm}x=2.29129$
$(-2.29129,1.24514);\hspace{1cm}(-2.29129,-1.24514)$
$(2.29129,-1.24514);\hspace{1cm}(2.29129,1.24514)$
