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) If m is the slope of the tangent line, then using the slope/point formula, the equation of the tangent line will be: y = m(x+1)+7 (b) Simultaneously solve the equation of the tangent line and ellipse to arrive at this quadratic in x: y=ax^2 + bx + c = 0 where a = ? b= ? c = ? by the quadratic then we can determine m is = ?

Explanation / Answer

y=mx-m-2

2*x^2+x*y+y^2=4

2*x^2+m*x^2-m*x-2*x+m^2*x^2+(m+2)^2-2*(m+2)*mx=4

x^2*(2+m+m^2)+x*(-m-2-2*m^2-4*m)+(m^2+4*m)=0

so comparing;

a=m^2+m+2

b=-2*m^2-5*m-2

c=m^2+4*m

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