Question

Let ab be a chord of the circle x2 + y2 = r2 subtending a right angle at the centre. Then the locus of the centroid of the triangle pab as p moves on the circle is

Answer 1

Step-by-step explanation:

Given Let ab be a chord of the circle x2 + y2 = r2 subtending a right angle at the centre. Then the locus of the centroid of the triangle pab as p moves on the circle is

• x^2 + y^2 = r^2  will be expression of circle
• Let co-ordinates of the point  p (x1,y1) = (r cos theta, r sin theta)
• coordinates of point A(x2,y2) and point B(x3,y3) will be (r,0) and (0,r)
• At centre chord AB traversed right angle at centre
• Now the coordinates of the centroid of triangle PAB is  
• (h,k) = (x1 + x2 + x3 / 3 , y1 + y2 + y3 / 3)
•         = (r cos theta + r + 0 /  3 , r sin theta + 0 + r / 3)
•       = 1/3(r + r cos theta)  1/3( r + r sin theta)
• Therefore h = 1/3 (r + r cos theta) -------------1
• Or h – 1/3 = 1/3 r cos theta
• Also P = 1/3 (r + r sin theta)
• Or P – 1/3 r = 1/3 r sin theta -------------------2
• Squaring and adding 1 and 2 we get
• (h – 1/3 r)^2 + (P – 1/3 r)^2
• = 1/9 r^2(cos^2 theta + sin^2 theta)
• = 1/9 r^2
• Therefore locus of centroid of triangle pad is                                                  (x – r/3)^2 + (y – r/3)^2 = r^2 / 9 is a circle.

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