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Geometry

Isometries

Translation

         

Point \((2, 1)\) is moved to point \((10,11)\) by \[T: (x, y) \rightarrow (x+a, y+b).\] What is \(a+b\)?

If we translate the point \((9, 3)\) by \((13, 2),\) what would be the new point?

The curve \(y=2x^2+3x-9\) undergoes a parallel translation by \(7\) units in the positive direction of the \(x\)-axis and \(-3\) units in the positive direction of the \(y\)-axis. If the equation of the resulting curve is \(y=ax^2+bx+c,\) what is \(a+b+c?\)

Consider a translation of axes \[T: (x, y) \to (x+a, y+b)\] such that the coordinates of a point \(P=(7, 4)\) with respect to the new system of coordinates after translation are \(P'=(10,-11).\) What is the value of \(a+b?\)

If the point \((m,n)\) is shifted by \(6\) in the positive \(x\) direction and by \(-12\) in the positive \(y\) direction, then it coincides with the center of the circle \[x^2+y^2-6x+2y-13=0.\] What is the value of \(m+n?\)

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