# intersecting point of two lines

The circle has R radius and and ellipse is intersecting the circle. I need to findout $x_c$ and $y_c$, which is the midpoint of the 2 intersected point of ellipse.Line 3 is the tangent of the ellipsoid which passes through the farthest point on ellipse.

The equation of line 1:

$$\frac{x-x_c}{x_c-0}=\frac{y-y_c}{y_c-0}\quad;\qquad y-y_c=\frac{y_c}{x_c}(x-x_c)$$

Perpendicular of line 1 is line 2, equation of line 2:

$$y-y_c=-\frac{y_c}{x_c}(x-x_c)$$

the equation of line 3:

$$ax+by+c=0$$

and parallel to this line and passing through the point $(x_c,y_c)$ is line 4. The equation of line 4:

$$y-y_c=-\frac ab(x-x_c)$$

I do not know the value of $x_c$ and $y_c$. How to find out these unknowns ($x_c$ and $y_c$).Are those equations correct? Any help will be appreciated.

• I don't understand your question, even after bringing it into a form which I can read more easily. What is the input from which you'd like to find $x_c$ and $y_c$? Does the circle of radius $R$ or the ellipse included in your illustration have anything to do with the question, since they aren't mentioned in the text? – MvG Dec 12 '14 at 8:19
• I also don't get it... I would expect something like "I need the value of $x_c$ and $y_c$" SUCH THAT.... – Umberto Dec 12 '14 at 8:41
• I have changed my post. Please let me if you need more information. (Circle is x^2+y^2=R^2 and ellipse: ax^2+by^2+2cxy+2dx+2ey+f=0 – sam Dec 13 '14 at 18:07