Dear student As OA⊥OB,where O0,0 is origin, hence∠AOB=90°⇒AB is diameter Angle in a semi circle is 90°Hence,⇒AB=2r=2aLet the coordinates of A be c,0 and that of B be 0,bBy distance formulaAB=c-02+0-b2=2a⇒c2+b2=4a2 1Let the coordinates of centroid be p,qp,q=0+c+03,0+0+b3⇒c=3p⇒b=3qSquaring and addingc2+b2=9p2+q29p2+q2=4a2 using 1Hence locus is :9x2+y2=4a2 Regards

Q1

Dear student

As OA ⊥ OB, where O (0, 0) is origin, hence ∠ AOB = 90 ° \Rightarrow AB is diameter [Angle in a semi circle is 90 °] Hence, \Rightarrow AB = 2 r = 2 a Let the coordinates of A be (c, 0) and that of B be (0, b) By distance formula AB = \sqrt{{(c - 0)}^{2} + {(0 - b)}^{2}} = 2 a \Rightarrow c^{2} + b^{2} = 4 a^{2} . . . (1) Let the coordinates of centroid be (p, q) (p, q) = (\frac{0 + c + 0}{3}, \frac{0 + 0 + b}{3}) \Rightarrow c = 3 p \Rightarrow b = 3 q Squaring and adding c^{2} + b^{2} = 9 (p^{2} + q^{2}) 9 (p^{2} + q^{2}) = 4 a^{2} [using (1)] Hence locus is : 9 (x^{2} + y^{2}) = 4 a^{2}

Regards

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