(A) Let the centroid of $\triangle OAB$ be $(x, y)$.Since $O = (0, 0)$,$A = (a \sec t, b \tan t)$,and $B = (-a \tan t, b \sec t)$,the coordinates of t

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(A) Let the centroid of $\triangle OAB$ be $(x, y)$.Since $O = (0, 0)$,$A = (a \sec t, b \tan t)$,and $B = (-a \tan t, b \sec t)$,the coordinates of the centroid are given by:$x = \frac{0 + a \sec t - a \tan t}{3} \Rightarrow 3x = a(\sec t - \tan t) \dots (i)$$y = \frac{0 + b \tan t + b \sec t}{3} \Rightarrow 3y = b(\tan t + \sec t) \dots (ii)$Multiplying equations $(i)$ and $(ii)$:$(3x)(3y) = a(\sec t - \tan t) \cdot b(\sec t + \tan t)$$9xy = ab(\sec^2 t - \tan^2 t)$Since $\sec^2 t - \tan^2 t = 1$,we get:$9xy = ab$.

Class 11 Mathematics 10-1.Circle and System of Circles Locus Related Problem

Medium

If $t$ is a parameter,$A = (a \sec t, b \tan t)$,$B = (-a \tan t, b \sec t)$,and $O = (0, 0)$,then the locus of the centroid of $\triangle OAB$ is:

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A
$9xy = ab$
B
$xy = 9ab$
C
$x^2 - 9y^2 = a^2 - b^2$
D
$x^2 - y^2 = \frac{1}{9}(a^2 - b^2)$

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10-1.Circle and System of Circles

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If $t$ is a parameter,$A = (a \sec t, b \tan t)$,$B = (-a \tan t, b \sec t)$,and $O = (0, 0)$,then the locus of the centroid of $\triangle OAB$ is:

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