The center of the circle is (1,1)

The radius of the circle stays the same. We just need to reflect the point (1,1) about the line y = -2x + 4

We first need to find the equation of a line that is perpendicular to y = -2x + 4 and that passes through (1,1)

Gradient of this line
= -1/-2
=½

Sub (1,1) into y = ½x + c ,

1 = ½(1) + c
c = ½

So the equation of the perpendicular is
y = ½x + ½

Next, find the coordinates of the intersection point of the two lines.

½x + ½ = -2x + 4
5/2 x = 7/2
x = 7/5
= 1.4

y = -2(7/5) + 4
= 6/5
= 1.2

So the coordinates are (1.4,1.2)


Now (1,1) and its reflection are equidistant to the line y = -2x + 4.

The midpoint of the line joining these two points is () since it is the foot of the perpendicular from both points to the line.

We let the coordinates of the reflected point be (a,b).

Using our formula to find midpoint
(x1 + x2)/2 = x(midpoint)
(y1 + y2)/2 = y(midpoint)


(1 + a)/2 = 1.4
1 + a = 2.8
a = 1.8

(1 + b)/2 = 1.2
1 + b = 2.4
b = 1.4


The coordinates of the reflected point are
(1.8,1.4)


Equation of reflected circle is :

(x - 1.8)² + (y - 1.4)² = 10

Takeaway from this question is that for any reflection done, the line joining the two points is always perpendicular to the line of reflection .