Use Implicit Differentiation to find equation of tangent line to ellipse 5x^2+xy+4y^2=40 at the point (−2,−2)

Question

Use implicit differentiation to find an equation of the tangent line to the ellipse defined by  5x^2+xy+4y^2=40 at the point (−2,−2)

Answer 1

5x²+xy+4y²=40

Take derivative both sides.

(d/dx)(5x²+xy+4y²)=0

10x+x(dy/dx)+y+8y(dy/dx)=0

(dy/dx)(x+8y)+10x+y=0

dy/dx=-(10x+y)/(x+8y)

dy/dx_(-2,-2) =-((10*-2)+(-2))/((-2)+8(-2))

               =+22/-18

               =-11/9

Calculate the tangent equation.

(y-(-2))=-( 11/9)(x-(-2))

y+2=-(11/9)(x+2)

9y+18=-11x-22

11x+9y+40=0