# Find the points of intersection of the curves : (y-1)^2 = x + 7 and x^2 + (y+5)^2 = 85.

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We have to find the points of intersection of (y-1)^2 = x + 7 and x^2 + (y+5)^2 = 85.

(y-1)^2 = x + 7

=> x = (y - 1)^2 - 7

substitute in x^2 + (y + 5)^2 = 85

((y - 1)^2 - 7)^2 + ( y + 5)^2 = 85

=> (y^2 + 1 - 2y - 7)^2 + y^2 + 25 + 10y = 85

=> (y^2 - 2y - 6)^2 + y^2 + 25 + 10y = 85

=> y^4 + 4y^2 + 36 - 4y^3 + 24y - 12y^2 + y^2 + 25 + 10y = 85

=> y^4 - 4y^3 - 7y^2 + 34y - 24 = 0

=> y^4 - 4y^3 - 7y^2 + 28y + 6y - 24 = 0

=> y^3(y - 4) - 7y(y - 4) + 6(y - 4) = 0

=> (y^3 - 7y + 6)(y - 4) = 0

=> (y^3 - 2y^2 + 2y^2 - 4y - 3y + 6)(y - 4) = 0

=> (y^2(y - 2) + 2y(y - 2) - 3(y - 2))(y - 4) = 0

=> (y - 4)(y - 2)(y^2 + 2y - 3) = 0

=> (y - 4)(y - 2)(y^2 + 3y - y - 3) = 0

=> (y - 4)(y - 2)(y(y + 3) - 1(y + 3)) = 0

=> (y - 4)(y - 2)(y - 1)(y + 3)

We get y = 4 , y = 2 , y = 1 and y = -3

As x = y^2 - 2y - 6, the corresponding values of x are:

x = 2 , x = -6 , -7 and x = 9

**The required points of intersection of the curves are ( 2,4), (-6,2), (-7 , 1) and ( 9, -3)**

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