If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than …

Question

If tangents are drawn to the ellipse x2  + 2y2  = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :

Accepted Answer

Equation of general tangent on ellipse

$${x \over {a\,\sec \theta }} + {y \over {b\cos ec\theta }} = 1$$

$$a = \sqrt 2 ,\,\,b = 1$$

$$ \Rightarrow {x \over {\sqrt 2 \sec \theta }} + {y \over {\cos ec\theta }} = 1$$

Let the midpoint be (h, k)

$$h = {{\sqrt 2 \sec \theta } \over 2} \Rightarrow \cos \theta  = {1 \over {\sqrt 2 h}}$$

and $$k = {{\cos ec\theta } \over 2} \Rightarrow \sin \theta  = {1 \over {2k}}$$

$$ \because $$  $${\sin ^2}\theta  + {\cos ^2}\theta  = 1$$

$$ \Rightarrow $$  $${1 \over {2{h^2}}} + {1 \over {4{k^2}}} = 1$$

$$ \Rightarrow $$  $${1 \over {2{x^2}}} + {1 \over {4{y^2}}} = 1$$

Solutions · 1