the locus of the mid point of the portion of the tangent to the ellipse( x^2/a^2)+(y^2/b^2)=1 included between the coordinate axis is

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$p a r a m e t r i c e q u a t i o n o f \tan g e n t t o e l l i p s e \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 i s \frac{x \cos θ}{a} + \frac{y \sin θ}{b} = 1 i t t o u c h e s t h e c o o r d i n a t e a x e s a t (a s e c θ, 0) a n d (0, b \cos e c θ) l e t t h e m i d p o i n t o f p o r t i o n o f \tan g e n t b e (h, k) s o, (h, k) = (a s e c θ / 2, b \cos e c θ / 2) w h i c h i s m i d p o i n t o f (a s e c θ, 0) a n d (0, b \cos e c θ) h = a s e c θ / 2 g i v e s \cos θ = a / 2 h k = b c o s e c θ / 2 g i v e s \sin θ = b / 2 k \cos^{2} θ + \sin^{2} θ = 1 {(\frac{a}{2 h})}^{2} + {(\frac{b}{2 k})}^{2} = 1 o r a^{2} k^{2} + b^{2} h^{2} = 4 h^{2} k^{2} s o l o c u s i s a^{2} y^{2} + b^{2} x^{2} = 4 x^{2} y^{2}$

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