


'''simple docstring'''






from math import isclose, sqrt






def 		_A (lowerCAmelCase__     :int ,  lowerCAmelCase__     :List[Any] ,  lowerCAmelCase__     :List[str]						)      ->     tuple[float, float, float]:




     '''simple docstring'''



     _a						=							point_y / 4 / point_x
     _a						=							2 * normal_gradient / (1 + normal_gradient * normal_gradient)
     _a						=							(1 - normal_gradient * normal_gradient) / (
         1 + normal_gradient * normal_gradient
     )
     _a						=							(sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

     # to find the next point, solve the simultaeneous equations:
     # y^2 + 4x^2 = 100
     # y - b = m * (x - a)
     # ==> A x^2 + B x + C = 0
     _a						=							outgoing_gradient**2 + 4
     _a						=							2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
     _a						=							(point_y - outgoing_gradient * point_x) ** 2 - 1_00

     _a						=							(
         -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term						)
     ) / (2 * quadratic_term)
     _a						=							(
         -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term						)
     ) / (2 * quadratic_term)

     # two solutions, one of which is our input point
     _a						=							x_minus if isclose(_UpperCamelCase ,  _UpperCamelCase						) else x_plus
     _a						=							point_y + outgoing_gradient * (next_x - point_x)

     return next_x, next_y, outgoing_gradient






def 		_A (lowerCAmelCase__     :List[Any] = 1.4 ,  lowerCAmelCase__     :List[Any] = -9.6						)      ->     int:




     '''simple docstring'''



     _a						=							0
     _a						=							first_x_coord
     _a						=							first_y_coord
     _a						=							(1_0.1 - point_y) / (0.0 - point_x)

     while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
          _a						=							next_point(_UpperCamelCase ,  _UpperCamelCase ,  _UpperCamelCase						)
          num_reflections += 1

     return num_reflections


if __name__ == "__main__":
 print(f'''{solution() = }''')








def      lowerCamelCase_						(				_UpperCamelCase ,				_UpperCamelCase )		->						str:






				"""simple docstring"""


				return "\n".join(
				    f'''{number} * {i} = {number * i}''' for i in range(1 ,				number_of_terms + 1 ) )


if __name__ == "__main__":
	print(multiplication_table(number=5, number_of_terms=1_0))
