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Projection of a Point on Conic and Distance of a Point from Conic Using Conic Intersection

1. Let's consider a Conic represented by General Quadratic Equations in 2 Variables as follows
   
   \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0\)   ...(1)
   
   The Projection of any given Point (\(x_0,y_0\)) on the Conic given by equation (1) above can be calculated using following steps
   1. Find the Slope of the Conic. Since the Conic is an Implicit Function of \(x\) and \(y\) (i.e. \(F(x,y)=0\)), it's Partial Derivatives with respect to Variables \(x\) and \(y\) are calculated as
      
      \({\Large \frac{\partial F}{\partial x}} = 2Ax + By + D\)   ...(2)
      
      \({\Large \frac{\partial F}{\partial y}} = 2Cy + Bx + E\)   ...(3)
      
      Hence, the Slope of the Conic at Any Point (\(x,y\)) is given as
      
      \(-{\Large \frac {\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}}=-{\Large \frac{2Ax + By + D}{2Cy + Bx + E}}\)   ...(4)
      
      
   2. Find the Slope of Any Line Connecting the Point (\(x_0,y_0\)) to any Point (\(x,y\)) on the Conic is given below
      
      Slope of Line \(={\Large \frac{y-y_0}{x-x_0}}\)
   3. For Any Point of Projection (\(x,y\)) on the Conic of Point (\(x_0,y_0\)), the Slope of the Line Connecting the Point (\(x_0,y_0\)) to Point (\(x,y\)) is Perpendicular to Slope of the Conic at Point (\(x,y\)). Hence,
      
      Slope of Conic \(\times\) Slope of Line  \( = -{\Large \frac{2Ax + By + D}{2Cy + Bx + E}} \times {\Large \frac{y-y_0}{x-x_0}}=-1\)
      
      \(\Rightarrow {\Large \frac{2Ax + By + D}{2Cy + Bx + E}} ={\Large \frac{x-x_0}{y-y_0}}\)
      
      \(\Rightarrow (2Cy + Bx + E) (x- x_0) - (2Ax + By + D)(y-y_0) = 0\)
      
      \(\Rightarrow Bx^2 - By^2 + (2C - 2A)xy + (2Ay_0 + E - Bx_0)x + (By_0 -D - 2Cx_0)y + Dy_0 - Ex_0=0\)   ...(5)
   4. Any Point of Projection (\(x,y\)) on the Conic of Point (\(x_0,y_0\)) Must Satisfy Both equations (1) and (5). Hence, the Any Point of Projection (\(x,y\)) on the Conic of Point (\(x_0,y_0\)) can be calculated as Real Point(s) of Intersection Between 2 Conics given by equations (1) and (5).
      
      However, if equation(1) represents a Circle, value of coefficient of \(B=0\) and \(A=C\), which makes the coefficients of \(x^2\), \(y^2\) and, \(xy\) in equation (5) \(0\). In such case the Points of Projection (\(x,y\)) on the Circle of Point (\(x_0,y_0\)) can be calculated as Point(s) of Intersection between Circle given by equation (1) and equation of Line given as follows
      
      \((2Ay_0 + E - Bx_0)x + (By_0 -D - 2Cx_0)y + Dy_0 - Ex_0=0\)   ...(6)
      
      Please note that the Line given by equation (6) above has same coefficients of \(x\) and \(y\) and the constant as of equation (5) above.
2. Once all the Points of Projection (\(x,y\)) have been calculated, their Distances can be calculated from the Point (\(x_0,y_0\)). The one with the Minimum Value of Distance is the Distance Between the Point and the Conic.