there are two lines through the given point, which are tangent to the given curve. Find the equation each of these lines.
4x^2- 5xy + 2y^2 + 3x - 2y = 0 , P ( 2,3)
need assistance on this please.
thank you
1. All lines passing through P have the equation: $\displaystyle y = m(x-2)+3$
2. Replace the term y in your equation of the ellipse by m(x-2)+3. You'll get a quadratic in x:
$\displaystyle x^2(2m^2 - 5m + 4) - x(8m^2 - 20·m + 12) + 4(2m^2 - 5·m + 3) = 0$
Solve for x.
3. Usually a straight line intercepts an ellipse in 2 or in 1 point or it is a passante that means there are no common points. The case that there exists only one point of interception occurs if the straight line is a tangent to the ellipse. This will happen if the discriminant equals zero:
$\displaystyle \sqrt{- 2·m^2 + 5·m - 3 - 2·m^2 + 5·m - 3)} = 0$
Solve for m. You should come out with $\displaystyle m = 1~\vee~m=\frac32$
4. Plug in these values into the equation of the line.