y = k ± b 1 − ( x − h ) 2 a 2 {\displaystyle y=k\pm b{\sqrt {1-{(x-h)^{2} \over a^{2}}}}\,}
m = ∓ b a x − h a 2 − ( x − h ) 2 {\displaystyle m=\mp {b \over a}{x-h \over {\sqrt {a^{2}-(x-h)^{2}}}}\,}
( x 1 , y 1 ) , m 1 ; ( x 2 , y 2 ) , m 2 {\displaystyle (x1,y1),m1;(x2,y2),m2\,}
∓ b = a m a 2 − ( x − h ) 2 x − h {\displaystyle \mp b=am{\dfrac {\sqrt {a^{2}-(x-h)^{2}}}{x-h}}\,}
∓ b = a m 1 a 2 − ( x 1 − h 1 ) 2 x 1 − h − ∓ b = a m 2 a 2 − ( x 2 − h 2 ) 2 x 2 − h {\displaystyle {\begin{aligned}\mp b&=am_{1}{\dfrac {\sqrt {a^{2}-(x_{1}-h_{1})^{2}}}{x_{1}-h}}\\-\quad \mp b&=am_{2}{\dfrac {\sqrt {a^{2}-(x_{2}-h_{2})^{2}}}{x_{2}-h}}\\\end{aligned}}\,}
a m 1 a 2 − ( x 1 − h ) 2 x 1 − h = a m 2 a 2 − ( x 2 − h ) 2 x 2 − h {\displaystyle am_{1}{\dfrac {\sqrt {a^{2}-(x_{1}-h)^{2}}}{x_{1}-h}}=am_{2}{\dfrac {\sqrt {a^{2}-(x_{2}-h)^{2}}}{x_{2}-h}}\,}
m 1 a 2 − ( x 1 − h ) 2 x 1 − h = m 2 a 2 − ( x 2 − h ) 2 x 2 − h {\displaystyle m_{1}{\dfrac {\sqrt {a^{2}-(x_{1}-h)^{2}}}{x_{1}-h}}=m_{2}{\dfrac {\sqrt {a^{2}-(x_{2}-h)^{2}}}{x_{2}-h}}\,}
m 1 2 a 2 − ( x 1 − h ) 2 ( x 1 − h ) 2 = m 2 2 a 2 − ( x 2 − h ) 2 ( x 2 − h ) 2 {\displaystyle m_{1}^{2}{\dfrac {a^{2}-(x_{1}-h)^{2}}{(x_{1}-h)^{2}}}=m_{2}^{2}{\dfrac {a^{2}-(x_{2}-h)^{2}}{(x_{2}-h)^{2}}}\,}
a 2 m 1 2 ( x 1 − h ) 2 − m 1 2 = a 2 m 2 2 ( x 2 − h ) 2 − m 2 2 {\displaystyle {\dfrac {a^{2}m_{1}^{2}}{(x_{1}-h)^{2}}}-m_{1}^{2}={\dfrac {a^{2}m_{2}^{2}}{(x_{2}-h)^{2}}}-m_{2}^{2}\,}
a 2 m 1 2 ( x 1 − h ) 2 − a 2 m 2 2 ( x 2 − h ) 2 = m 1 2 − m 2 2 {\displaystyle {\dfrac {a^{2}m_{1}^{2}}{(x_{1}-h)^{2}}}-{\dfrac {a^{2}m_{2}^{2}}{(x_{2}-h)^{2}}}=m_{1}^{2}-m_{2}^{2}\,}
a 2 m 1 2 ( x 2 − h ) 2 − a 2 m 2 2 ( x 1 − h ) 2 ( x 1 − h ) 2 ( x 2 − h ) 2 = m 1 2 − m 2 2 {\displaystyle {\dfrac {a^{2}m_{1}^{2}(x_{2}-h)^{2}-a^{2}m_{2}^{2}(x_{1}-h)^{2}}{(x_{1}-h)^{2}(x_{2}-h)^{2}}}=m_{1}^{2}-m_{2}^{2}\,}
a 2 m 1 2 ( x 2 2 − 2 x 2 h + h 2 ) − a 2 m 2 2 ( x 1 2 − 2 x 1 h + h 2 ) ( x 1 2 − 2 x 1 h + h 2 ) ( x 2 2 − 2 x 2 h + h 2 ) = m 1 2 − m 2 2 {\displaystyle {\dfrac {a^{2}m_{1}^{2}(x_{2}^{2}-2x_{2}h+h^{2})-a^{2}m_{2}^{2}(x_{1}^{2}-2x_{1}h+h^{2})}{(x_{1}^{2}-2x_{1}h+h^{2})(x_{2}^{2}-2x_{2}h+h^{2})}}=m_{1}^{2}-m_{2}^{2}\,}
0 = a 2 m 1 2 ( h 2 − 2 x 2 h + x 2 2 ) − a 2 m 2 2 ( h 2 − 2 x 1 h + x 1 2 ) {\displaystyle 0=a^{2}m_{1}^{2}(h^{2}-2x_{2}h+x_{2}^{2})-a^{2}m_{2}^{2}(h^{2}-2x_{1}h+x_{1}^{2})\,} − ( m 1 2 − m 2 2 ) [ h 4 − 2 ( x 1 + x 2 ) h 3 + 4 x 1 x 2 h 2 − 2 x 1 x 2 ( x 1 + x 2 ) h + x 1 2 x 2 2 ] {\displaystyle -(m_{1}^{2}-m_{2}^{2})[h^{4}-2(x_{1}+x_{2})h^{3}+4x_{1}x_{2}h^{2}-2x_{1}x_{2}(x_{1}+x_{2})h+x_{1}^{2}x_{2}^{2}]\,}
0 = [ − ( m 1 2 − m 2 2 ) ] h 4 + [ 2 ( m 1 2 − m 2 2 ) ( x 1 + x 2 ) ] h 3 + [ ( m 1 2 − m 2 2 ) ( a 2 − 4 x 1 x 2 ) ] h 2 {\displaystyle 0=[-(m_{1}^{2}-m_{2}^{2})]h^{4}+[2(m_{1}^{2}-m_{2}^{2})(x_{1}+x_{2})]h^{3}+[(m_{1}^{2}-m_{2}^{2})(a^{2}-4x_{1}x_{2})]h^{2}\,} + [ 2 x 1 x 2 ( m 1 2 − m 2 2 ) ( x 1 + x 2 ) + 2 a 2 ( m 1 2 x 2 − m 2 2 x 1 ) ] h + [ a 2 ( m 1 2 x 2 2 − m 2 2 x 1 2 ) + x 1 2 x 2 2 ( m 1 2 − m 2 2 ) ] {\displaystyle +[2x_{1}x_{2}(m_{1}^{2}-m_{2}^{2})(x_{1}+x_{2})+2a^{2}(m_{1}^{2}x_{2}-m_{2}^{2}x_{1})]h+[a^{2}(m_{1}^{2}x_{2}^{2}-m_{2}^{2}x_{1}^{2})+x_{1}^{2}x_{2}^{2}(m_{1}^{2}-m_{2}^{2})]\,}
0 = [ 1 ] h 4 + [ − 2 ( x 1 + x 2 ) ] h 3 + [ 4 x 1 x 2 − a 2 ] h 2 {\displaystyle 0=[1]h^{4}+[-2(x_{1}+x_{2})]h^{3}+[4x_{1}x_{2}-a^{2}]h^{2}\,} + [ − 2 x 1 x 2 ( m 1 2 − m 2 2 ) ( x 1 + x 2 ) + 2 a 2 ( m 1 2 x 2 − m 2 2 x 1 ) m 1 2 − m 2 2 ] h {\displaystyle +\left[-{\dfrac {2x_{1}x_{2}(m_{1}^{2}-m_{2}^{2})(x_{1}+x_{2})+2a^{2}(m_{1}^{2}x_{2}-m_{2}^{2}x_{1})}{m_{1}^{2}-m_{2}^{2}}}\right]h\,} + [ − a 2 ( m 1 2 x 2 2 − m 2 2 x 1 2 ) − x 1 2 x 2 2 ( m 1 2 − m 2 2 ) m 1 2 − m 2 2 ] {\displaystyle +\left[{\dfrac {-a^{2}(m_{1}^{2}x_{2}^{2}-m_{2}^{2}x_{1}^{2})-x_{1}^{2}x_{2}^{2}(m_{1}^{2}-m_{2}^{2})}{m_{1}^{2}-m_{2}^{2}}}\right]\,}
![{\displaystyle -(m_{1}^{2}-m_{2}^{2})[h^{4}-2(x_{1}+x_{2})h^{3}+4x_{1}x_{2}h^{2}-2x_{1}x_{2}(x_{1}+x_{2})h+x_{1}^{2}x_{2}^{2}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d04e4119becce2e6a307e7b7ae12a79a6a1008d)
![{\displaystyle 0=[-(m_{1}^{2}-m_{2}^{2})]h^{4}+[2(m_{1}^{2}-m_{2}^{2})(x_{1}+x_{2})]h^{3}+[(m_{1}^{2}-m_{2}^{2})(a^{2}-4x_{1}x_{2})]h^{2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1ea519658947b8dc94c9382d97363e690a9ef6)
![{\displaystyle +[2x_{1}x_{2}(m_{1}^{2}-m_{2}^{2})(x_{1}+x_{2})+2a^{2}(m_{1}^{2}x_{2}-m_{2}^{2}x_{1})]h+[a^{2}(m_{1}^{2}x_{2}^{2}-m_{2}^{2}x_{1}^{2})+x_{1}^{2}x_{2}^{2}(m_{1}^{2}-m_{2}^{2})]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef8d369db9a9738a1213464c0d0b371cea2a8211)
![{\displaystyle 0=[1]h^{4}+[-2(x_{1}+x_{2})]h^{3}+[4x_{1}x_{2}-a^{2}]h^{2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e222f7432c99c74615888f553ebcc97574bfb7c5)
![{\displaystyle +\left[-{\dfrac {2x_{1}x_{2}(m_{1}^{2}-m_{2}^{2})(x_{1}+x_{2})+2a^{2}(m_{1}^{2}x_{2}-m_{2}^{2}x_{1})}{m_{1}^{2}-m_{2}^{2}}}\right]h\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0852d1b9e2d0635c2ca203501a562075418675d2)
![{\displaystyle +\left[{\dfrac {-a^{2}(m_{1}^{2}x_{2}^{2}-m_{2}^{2}x_{1}^{2})-x_{1}^{2}x_{2}^{2}(m_{1}^{2}-m_{2}^{2})}{m_{1}^{2}-m_{2}^{2}}}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dfe0b57d0db204c2292018434a492fcf4a4d997)
