Solve Tangent Equation: Find A, B, & C for Ellipse

[kidia]

Hi any idea on this please.

Find the values of constant A,B and C if the ellipse 4x^2 +y^2+Ax+By+C=0 is to be tangent to the x-axis at the origin and to pass through the point P(-1,2)

 

[mathmike]

what i would do first is to put the equation into a form more readable as far as ellipces aree concerned

 

[Timbuqtu]

You can also do it straight ahaid:

(1) The ellipse must pass through (0,0). What does this tell you about C?

(2) The ellipse must be tangent to the x-axis too. So the normal to the ellipse can not have an x-component. Recall that the gradient of a function is perpendicular to it's constant value curves. So the normal to the ellipse at the origin is given by the gradient of $$4x^2 +y^2+A x+B y+C$$ at (0,0). This gives us A.

(3) Just plug in the point P to find the remaining unknown B.

 

[Maxos]

Dini's theorem: $$g(x_0, y_0)=0$$ and $$\nabla g|_{(x_0, y_0)} \neq \vec 0$$, then

$$\exists h(x) : g(x, h(x)) = 0$$ and

$$h'(x)=\frac {g_x} {g_y}$$

 

[kidia]

I didn`t got u on getting constant Value C Timbuqtu

 

[TD]

I didn`t got u on getting constant Value C Timbuqtu

Fill in the point (0,0) and you'll find immediately that c = 0.
If not, the ellipse wouldn't go through the origin.