find the min area of the triangle formed by the tangents to the ellipse x^2/a^2 +y^2/b^2 =1and the coordinate axis

eq of tangent to ellipse is given by

y = mx +(-) (a²m²+b²)^1/2

it cuts the coordinate axis at  [ 0, (a²m²+b²)^1/2  ]     &      [ (a²m²+b²)^1/2/m , 0 ]

area formed by these points & origin is

A = (1/2) [ (a²m²+b²)/m ]

now using maxima minima concept

diffenentiating this eq wrt m & after putting it  to 0 we get

m²a² - b² = 0

  m = +(-) (b/a)

so , area can be

A = (2b²)a/2b = ab                ( at m = b/a)

this is the minimum area of the triangle formed by tangent and coordinate axis....

Approved