
The equation of the Ellipse is written as:
b ² x ² – a ² y ² = – a ² b ² , where x = w / 2 and y = b + h Therefore : b ² x ² = a ² y ² – a ² b ² Collecting terms : b ² x ² = a ² ( y ² – b ² ) Dividing both sides of the equation by ( y ² – b ² ) : a ² = b ² x ² / ( y ² – b ² ) Taking the square root of both sides of the equation : a = bx / Ö ( y ² – b ² ) , and the axis on the xaxis = 2a 
The equation of the Ellipse is written as:
b ² x ² – a ² y ² = – a ² b ² , where x = w / 2 and b = y – h Transposing the term from the right side : a ² y ² – b ² x ² – a ² b ² = 0 Collecting like terms : a ² y ² – b ² ( a ² + x ² ) = 0 Dividing by ( a ² + x ² ) : y ² a ² / ( a ² + x ² ) – b ² = 0 Substituting for b to express the equation in terms of y and h : y ² a ² / ( a ² + x ² ) – ( y – h ) ² = 0 Expanding the term ( y – h ) ² : y ² a ² / ( a ² + x ² ) – ( y ² – 2 y h + h ² ) = 0 Removing the parentheses : y ² a ² / ( a ² + x ² ) – y ² + 2 y h – h ² = 0 Collecting terms : y ² [ a ² / ( a ² + x ² ) – 1 ] + y 2 h – h ² = 0 The equation is quadratic in y, where : A = a ² / ( a ² + x ² ) – 1 B = 2 h C = – h ² Substituting in the General Quadratic Equation : [ – B ± Ö ( B ² – 4 A C ) ] / 2A returns the value of y Therefore b = y – h , and the axis on the yaxis = 2b Although only one value for b is returned by the calculator, due to the ± sign in the General Quadratic Equation there are two possible solutions. The calculator returns values based on the negative value of the discriminant. 
The equation of the Ellipse is written as:
a ² x ² – a ² y ² = – a ² a ² , where x = w / 2 and y = a + h Therefore : a ² = y ² – x ² Substituting for x and y : a ² = ( a + h ) ² – (w / 2 ) ² Expanding the terms : a ² = a ² + 2 a h + h ² – w ² / 4 Transposing terms and multiplying all terms by 4 : 4 a h = w ² – 4 h ² Dividing both sides by 4 h : a = ( w ² – 4 h ² ) / 4 h 