A circle of variable radius given by x 2 + y 2 + 2gx + 2fy + c = 0 cuts the rectangular hyperbola x 2 – y 2 = 9a 2 in points P, Q, R and S at (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ) and (x 4 , y 4 ) then their centre of mean lies at the mid point of line joining their centres.
= – 
, 
= – 
(i) The centroid of the Δ PQR is :
Text Solution
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Ans.
(i)
Sol. Let (h, k)be the centroid of Δ PQR. Then x 1 + x 2 + x 3
= 3h and y 1 + y 2 + y 3 = 3k.
∴ 3h + x 4 = – 2g and 3k + y 4 = –2f
⇒ x 4 = –2g – 3h and y 4 = – 2 f – 3k
∴ Centroid of Δ is (–2g – 3h, –2 f – 3k)
(ii)
Sol. (x 4 , y 4 ) lies on x 2 – y 2 = a 2 .
∴ 
– 
= a 2 .
⇒ (–2g – 3h) 2 – (–2 f – 3k) 2 = a 2 .
∴ locus of (h, k) is,
(3x + 2g) 2 – (3y + 2 f ) 2 = a 2 .
or 
– 
= 
(iii)
Sol. Centre coordinates are 
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