The graph of the equation 2x2 + xy + y2 = 4 is the tilted ellipse pictured below

Question

The graph of the equation 2x2 + xy + y2 = 4 is the tilted ellipse pictured below; i.e. the points (x,y) in the plane that satisfy the equation yield the pictured ellipse. The point P=(1,-2) is on the ellipse and the tangent through P is also pictured.

a.)Ifmis the slope of the tangent line, then using the slope/point formula, the equation of the tangent line will be: y=m(x-_?_) +_?_

b)Simultaneously solve the equation of the tangent line and ellipse to arrive at this quadratic in x:

ax2+bx+c= 0 a=? b=? c=?

c)Use the reasoning in class and the quadratic formula to determine thatm=

Explanation / Answer

2x2 + xy + y2 = 4 4x+xdy/dx +2ydy/dx =0 dy/dx = -4x/ (x+2y) =>at 1,-2 we have dy/dx = 4/3 so y= mx+c =>y =4/3x+c C= -10/3

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