4) Find the image of the lircle \( x^{2}+y^{2}+2 x+2 y=2 \) after Clockulise rotation through \( 90^{\circ} \) about \( (-1,2) \)

To begin, let's rewrite the equation of the circle in standard form. The given equation is \( x^{2} + y^{2} + 2x + 2y - 2 = 0 \). Completing the square, we have \( (x+1)^{2} + (y+1)^{2} = 4 \). This represents a circle centered at \( (-1, -1) \) with a radius of \( 2 \). Now, for the 90° clockwise rotation around the point \( (-1, 2) \), we first translate the center of the circle to the origin by shifting it down from \( (-1, 2) \) to \( (0, 0) \). The center of the circle relative to the rotation point is \( (0, -3) \). Using the rotation matrix for a 90° clockwise rotation: \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \] This results in a transformation where \( x' = y \) and \( y' = -x \). Now applying this transformation to the center \( (0, -3) \): - New x-coordinate: \( y = -3 \) - New y-coordinate: \( -x = 0 \) The new center after rotation is \( (-3, 0) \). We then translate the center back to \( (-1, 2) \), resulting in: \[ (-3 + (-1), 0 + 2) = (-4, 2) \] Thus, the image of the circle after a 90° clockwise rotation around the point \( (-1, 2) \) is centered at \( (-4, 2) \) with the same radius \( 2 \). The equation of the new circle is: \[ (x + 4)^{2} + (y - 2)^{2} = 4 \] And there you have it! Happy circling!