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Show that the equation of the tangent to...

Show that the equation of the tangent to the ellipse `(x^(2))/(a^(2))+ (y^(2))/(b^(2)) = 1 " at " (x_(1), y_(1)) " is " ("xx"_(1))/(a^(2)) + ("yy"_(1))/(b^(2)) = 1`

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`(x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 rArr (2x)/(a^(2)) + (2y)/(b^(2)).(dy)/(dx) = 0` [on differentiating w.r.t. x]
`(dy)/(dx) = (-b^(2)x)/(a^(2)y) rArr ((dy)/(dx))_((x_(1), y_(1)) = (-b^(2) x_(1))/(a^(2) y_(1))`
So, the equation of the tangent at `(x_(1), y_(1))` is
`(y - y_(1))/(x -x_(1)) = (-b^(2)x_(1))/(a^(2) y_(1)) rArr b^(2) "xx"_(1) + a^(2) yy_(1) = b^(2) x_(1)^(2) + a^(2) y_(1)^(2)`
On dividing throughout by `a^(2) b^(2)`, we get `("xx"_(1))/(a^(2)) + (yy_(1))/(b^(2)) = (x_(1)^(2))/(a^(2)) + (y_(1)^(2))/(b^(2))`,
i.e., `("xx"_(1))/(a^(2)) + (yy_(1))/(b^(2)) = 1 [ :' (x_(1), y_(1)) " lies on " (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 rArr (x_(1)^(2))/(a^(2)) + (y_(1)^(2))/(b^(2)) = 1]`
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