translate circle given as x^2+y^2+4x-8y+4=o using T(x-4,y-6)
x^2+y^2+4x-8y+4=0
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(x-4)^2 + (y-6)^2 + 4(x-4) - 8(y-6) + 4 = 0
leaving the expanding and simplification up to you
What is the rule for the given transportation
To translate a circle, we need to shift its center by a certain amount in both the x and y directions. In this case, we need to shift the circle by (4, 6) in order to match the new coordinates T(x-4,y-6).
Let's break down the process step by step:
1. Write the equation of the original circle:
x^2 + y^2 + 4x - 8y + 4 = 0
2. Substitute the new coordinates T(x-4, y-6) for (x, y):
(x-4)^2 + (y-6)^2 + 4(x-4) - 8(y-6) + 4 = 0
3. Simplify and expand the equation:
(x^2 - 8x + 16) + (y^2 - 12y + 36) + 4x - 16 - 8y + 48 + 4 = 0
4. Combine like terms:
x^2 - 8x + 4x + y^2 - 12y - 8y + 16 + 36 + 48 + 4 - 16 = 0
5. Simplify further:
x^2 - 4x + y^2 - 20y + 72 = 0
Therefore, the translated equation of the circle, when shifted by (4, 6), is:
x^2 - 4x + y^2 - 20y + 72 = 0