Point(s) of Intersection Between a Line and a Conic

1. A Line is mathematically represented by a Linear Equation in 2 Variables as follows
   
   \(A_Lx + B_Ly + C_L=0\)   ...(1)
2. A Conic is represented by a General Quadratic Equation in 2 Variables as follows
   
   \(A_Cx^2 + B_Cxy + C_Cy^2 + D_Cx + E_Cy + F_C=0\)   ...(2)
3. A Line and a Conic can Intersect in either 1 or 2 Points or Not Intersect at all. The Point(s) of Intersection Between a Line and a Conic can be found out using the following steps
   1. Convert Linear Equation as given in (1) to Explicit Form where either the variable \(x\) is given in terms of equation of variable \(y\) (as given in equation (3) below) or the variable \(y\) is given in terms of equation of variable \(x\) (as given in equation (4) below)
      
      \(x=-\frac{B_L}{A_L}y - \frac{C_L}{A_L}\)   ...(3)
      
      \(y=-\frac{A_L}{B_L}x - \frac{C_L}{B_L}\)   ...(4)
   2. Put the value of \(x\) or \(y\) thus obtained from equation (3) and (4) in the Equation of Conic given by equation (2) to obtain a Quadratic Equation in 1 Variable.
      
      The Quadratic Equation in terms of Variable \(y\) can be obtained by putting the value of Varible \(x\) from equation (3) above in equation (2) as follows
      
      \(A_Cx^2 + B_Cxy + C_Cy^2 + D_Cx + E_Cy + F_C=0\)
      
      \(\Rightarrow A_C{(-\frac{B_L}{A_L}y - \frac{C_L}{A_L})}^2 + B_C(-\frac{B_L}{A_L}y - \frac{C_L}{A_L})y + C_Cy^2 + D_C(-\frac{B_L}{A_L}y - \frac{C_L}{A_L}) + E_Cy + F_C=0\)
      
      \(\Rightarrow A_C(\frac{{B_L}^2}{{A_L}^2}y^2 + \frac{{C_L}^2}{{A_L}^2} + 2\frac{B_LC_L}{{A_L}^2}y) + B_C(-\frac{B_L}{A_L}y^2 - \frac{C_L}{A_L}y) + C_Cy^2 + D_C(-\frac{B_L}{A_L}y - \frac{C_L}{A_L}) + E_Cy + F_C=0\)
      
      \(\Rightarrow \frac{A_C{B_L}^2}{{A_L}^2}y^2 + \frac{A_C{C_L}^2}{{A_L}^2} + 2\frac{A_CB_LC_L}{{A_L}^2}y -\frac{B_CB_L}{A_L}y^2 - \frac{B_CC_L}{A_L}y + C_Cy^2 -\frac{D_CB_L}{A_L}y - \frac{D_CC_L}{A_L} + E_Cy + F_C=0\)
      
      \(\Rightarrow (\frac{A_C{B_L}^2}{{A_L}^2} -\frac{B_CB_L}{A_L} + C_C) y^2 + (2\frac{A_CB_LC_L}{{A_L}^2} - \frac{B_CC_L}{A_L} - \frac{D_CB_L}{A_L} + E_C )y + \frac{A_C{C_L}^2}{{A_L}^2} - \frac{D_CC_L}{A_L} + F_C=0\)
      
      \(\Rightarrow (\frac{A_C{B_L}^2 -B_CB_LA_L + C_C{A_L}^2}{{A_L}^2}) y^2 + (\frac{2A_CB_LC_L - B_CC_LA_L - D_CB_LA_L + E_C{A_L}^2}{{A_L}^2})y + \frac{A_C{C_L}^2 - D_CC_LA_L + F_C{A_L}^2}{{A_L}^2}=0\)
      
      \(\Rightarrow (A_C{B_L}^2 -B_CB_LA_L + C_C{A_L}^2) y^2 + (2A_CB_LC_L + E_C{A_L}^2 - B_CC_LA_L - D_CB_LA_L)y + A_C{C_L}^2 - D_CC_LA_L + F_C{A_L}^2=0\)   ...(5)
      
      Similarly, the Quadratic Equation in terms of Variable \(x\) can be obtained by putting the value of Variable \(y\) from equation (4) above in equation (2) as follows
      
      \(A_Cx^2 + B_Cxy + C_Cy^2 + D_Cx + E_Cy + F_C=0\)
      
      \(\Rightarrow A_Cx^2 + B_Cx(-\frac{A_L}{B_L}x - \frac{C_L}{B_L}) + C_C{(-\frac{A_L}{B_L}x - \frac{C_L}{B_L})}^2 + D_Cx + E_C(-\frac{A_L}{B_L}x - \frac{C_L}{B_L}) + F_C=0\)
      
      \(\Rightarrow A_Cx^2 + B_C(-\frac{A_L}{B_L}x^2 - \frac{C_L}{B_L}x) + C_C(\frac{{A_L}^2}{{B_L}^2}x^2 + \frac{{C_L}^2}{{B_L}^2} + 2\frac{A_LC_L}{{B_L}^2}x) + D_Cx + E_C(-\frac{A_L}{B_L}x - \frac{C_L}{B_L}) + F_C=0\)
      
      \(\Rightarrow A_Cx^2 -\frac{B_CA_L}{B_L}x^2 - \frac{B_CC_L}{B_L}x + \frac{C_C{A_L}^2}{{B_L}^2}x^2 + \frac{C_C{C_L}^2}{{B_L}^2} + 2\frac{C_CA_LC_L}{{B_L}^2}x + D_Cx -\frac{E_CA_L}{B_L}x - \frac{E_CC_L}{B_L} + F_C=0\)
      
      \(\Rightarrow (A_C -\frac{B_CA_L}{B_L}+ \frac{C_C{A_L}^2}{{B_L}^2})x^2 +(- \frac{B_CC_L}{B_L} + 2\frac{C_CA_LC_L}{{B_L}^2} + D_C -\frac{E_CA_L}{B_L})x + \frac{C_C{C_L}^2}{{B_L}^2} - \frac{E_CC_L}{B_L} + F_C=0\)
      
      \(\Rightarrow (\frac{A_C{B_L}^2 - B_CA_LB_L + C_C{A_L}^2}{{B_L}^2})x^2 +(\frac{-B_CC_LB_L + 2C_CA_LC_L + D_C{B_L}^2 - E_CA_LB_L}{{B_L}^2})x + \frac{C_C{C_L}^2 - E_CC_LB_L + F_C{B_L}^2}{{B_L}^2}=0\)
      
      \(\Rightarrow (A_C{B_L}^2 - B_CA_LB_L + C_C{A_L}^2)x^2 +(2C_CA_LC_L + D_C{B_L}^2 - B_CC_LB_L - E_CA_LB_L)x + C_C{C_L}^2 - E_CC_LB_L + F_C{B_L}^2=0\)   ...(6)
   3. Solve the Quadratic Equation (5) using Quadratic Formula to obtain the values of Variable \(y\) as follows
      
      \(y=\frac{-(2A_CB_LC_L + E_C{A_L}^2 - B_CC_LA_L - D_CB_LA_L) \pm \sqrt{{(2A_CB_LC_L + E_C{A_L}^2 - B_CC_LA_L - D_CB_LA_L)}^2 - 4 (A_C{B_L}^2 -B_CB_LA_L + C_C{A_L}^2)(A_C{C_L}^2 - D_CC_LA_L + F_C{A_L}^2) }}{2(A_C{B_L}^2 -B_CB_LA_L + C_C{A_L}^2)}\)   ...(7)
      
      Similarly, solve the Quadratic Equation (6) using Quadratic Formula to obtain the values of Variable \(x\) as follows
      
      \(x=\frac{-(2C_CA_LC_L + D_C{B_L}^2 - B_CC_LB_L - E_CA_LB_L) \pm \sqrt{{(2C_CA_LC_L + D_C{B_L}^2 - B_CC_LB_L - E_CA_LB_L)}^2 -4 (A_C{B_L}^2 - B_CA_LB_L + C_C{A_L}^2) (C_C{C_L}^2 - E_CC_LB_L + F_C{B_L}^2) } }{2(A_C{B_L}^2 - B_CA_LB_L + C_C{A_L}^2)}\)   ...(8)
      
      If the values of \(x\) and \(y\) obtained from equations (7) and (8) are Real then the Line Intersects the Conic. If the values of \(x\) and \(y\) obtained from equations (7) and (8) are Complex or Imaginary then the Line Does Not Intersect the Conic.
   4. If the values of \(x\) and \(y\) obtained from equations (7) and (8) are Real then the Point(s) of Intersection can be found out by putting the values of \(y\) from equation (7) in equation (3). Alternatively, the Point(s) of Intersection can be found out by putting the values of \(x\) from equation (8) in equation (4).