Three vertices of each triangle on the contact ellipse ellipse called triangle, known point \ (A \) is a minor axis end point of an ellipse, if \ (A \) ellipse vertex right angle isosceles right contact and only three triangles, ellipses the heart rate is in the range from \ (\ underline {\ qquad \ qquad} \) .
analytical: \ (\ triangle the ABC \) and elliptical as shown, may assume elliptic equation is \ [\ dfrac {x ^ 2 } {a ^ 2} + \ dfrac {y ^ 2} {b ^ 2} = 1, a> b> 0. \]
Thus the original elliptic rectangular coordinate equations in the new coordinate system is \ [\ dfrac {x ^ 2 } {a ^ 2} + \ dfrac {y ^ 2} {b ^ 2} + \ dfrac {2y} {b} = 0. \ qquad (\ ast) \ ] to establish a new coordinate origin coordinates \ (O \) of the pole, with a new coordinate system \ (X \) n axle polar axis is a polar coordinate system, set \ [\ angle CAx = \ theta, \ theta \ in \ left (0, \ dfrac {\ pi} {2} \ right). \] is \ (C, B \) point can be set polar \ [C \ left (\ rho_1,2 \ pi- \ theta \ right), B \ left (\ rho_2,2 \ pi- \ theta- \ dfrac {\ pi} {2} \ right), \] is \ (C \ ) Cartesian coordinate can be expressed as \ [C \ left (\ rho_1 \ cos \ theta, - \ rho_1 \ sin \ theta \ right), B \ left (- \ rho_2 \ sin \ theta, - \ rho_2 \ cos \ theta \ right) \] the \ (C, B \) two coordinates into equation \ ((\ AST) \) , and organize available \ [\ Rho_1 = \ dfrac { \ dfrac {2 \ sin \ theta} {b}} {\ dfrac {\ cos ^ 2 \ theta} {a ^ 2} + \ dfrac {\ sin ^ 2 \ theta} {b ^ 2}}, \ rho_2 = \ dfrac {\ dfrac {2 \ cos \ theta} {b}} {\ dfrac {\ sin ^ 2 \ theta} {a ^ 2} + \ dfrac {\ cos ^ 2 \ theta } {b ^ 2}}. \] binding \ (\ rho_1 = \ rho_2 \ ) can be obtained on \ (\ Theta \) equation \ [\ dfrac {b ^ 2 } {a ^ 2} \ tan ^ 3 \ theta- \ tan ^ 2 \ theta + \ tan \ theta- \ dfrac {b ^ 2} {a ^ 2} = 0, \ theta \ in \ left (0, \ dfrac {\ pi} {2} \ right). \]
questions regarding meaning \ (\ tan \ theta \) equation has three solutions, easy to obtain \ (\ dfrac {b ^ 2 } {a ^ 2} \) ranges \ (\ left (0 , \ dfrac. 1 {{}}. 3 \ right) \) , and therefore the range of eccentricity is ask \ (\ left (\ dfrac {\ sqrt {} {}. 6. 3},. 1 \ right) \) .