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Q1. Show that for all values of p, the circle x2 + y2 - x(3p+4) –y(p-2) + 10p =0 passes through the point (3,1). If p varies, find the locus of the centre of the above circle.
Q2. Whatever be the values of θ, prove that the locus of the point of intersection of the straight lines y =atanθ and asin3θ + ycos3θ = asinθcosθ is a circle. Find the equation of the circle.
Q3. Prove that the square of the distance between the two points (x1 , y1) and (x2 , y2) of the circle x2 +y2 = a2 is 2(a2-x1x2-y1y2).
Q1:
put x=3, y=1 in the eq. U get 9+1-3(3p+4)-(p-2)+10p=12-12-10p+10p=0.
ie. For any value of p the pt.(3,1) satisfy the eq. Hence proved.
Let center be (h,k). From eq.
h=(3p+4)/2 => p=(2h-4)/3
k=(p-2)/2 . Sub. value of p we get:
3k-h+5=0
ie. locus of center is:
3y-x+5=0 ie. straight line
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