Question

circle a and circle b are both centered at the origin. circle a has radius r and circle b has radius s. which transformation maps circle a onto circle b?
(x, y) → (rx, ry)
(x, y) → (sx, sy)
(x, y) → (\frac{rx}{s}, \frac{ry}{s})
(x, y) → (\frac{sx}{r}, \frac{sy}{r})

Explanation:

Step1: Recall the transformation of scaling a circle

When we want to map a circle \(A\) with radius \(r\) to a circle \(B\) with radius \(s\), we use a similarity transformation. A similarity transformation for circles (which are similar figures) in the coordinate - plane (centered at the origin \((0,0)\)) is a dilation. If the center of the circle is \((0,0)\), and we have a point \((x,y)\) on circle \(A\), after dilation, the point on circle \(B\) will be \((\frac{s}{r}x,\frac{s}{r}y)\).

Let's check each option:

• Option \((x,y)\to(rx,ry)\): This would scale a circle with radius \(1\) (if we assume a unit - circle as a reference) to a circle with radius \(r\), which is the wrong direction if we want to map from radius \(r\) to radius \(s\).
• Option \((x,y)\to(sx,sy)\): If we assume a unit - circle (\(r = 1\)), this would map to a circle of radius \(s\). But if the original circle has radius \(r

eq1\), this is incorrect.

• Option \((x,y)\to(\frac{x}{s},\frac{y}{s})\): This is a contraction (if \(s>1\)) or expansion (if \(s < 1\)) in the wrong proportion. For example, if the original radius is \(r\) and we want to get to radius \(s\), this does not use the ratio \(\frac{s}{r}\).
• Option \((x,y)\to(\frac{sx}{r},\frac{sy}{r})\):

Let the equation of circle \(A\) be \(x^{2}+y^{2}=r^{2}\) (since it is centered at the origin \((0,0)\) with radius \(r\)). After the transformation \((x,y)\to(\frac{sx}{r},\frac{sy}{r})\), we substitute \(x'=\frac{sx}{r}\) and \(y'=\frac{sy}{r}\) into the equation of a circle. Then \(x=\frac{rx'}{s}\) and \(y = \frac{ry'}{s}\). Substituting into \(x^{2}+y^{2}=r^{2}\), we get \((\frac{rx'}{s})^{2}+(\frac{ry'}{s})^{2}=r^{2}\). Simplifying, \(\frac{r^{2}x'^{2}+r^{2}y'^{2}}{s^{2}}=r^{2}\), and then \(x'^{2}+y'^{2}=s^{2}\), which is the equation of a circle centered at the origin with radius \(s\).

Answer:

\((x,y)\to(\frac{sx}{r},\frac{sy}{r})\)