Let $K$ be a circle $(x-1)^2+(y-4)^2=5$ and $P$ a line with the equation $y=-3x$. We transform the circle $K$ with a transformation, defined with the next conditions:
points are tranformed, so that their distance from the line $P$ gets multiplied by the factor $3$
vector from the original point to the image is perpendicular to $P$
- the original points and their images are on the same side of $P$
Let $E$ be the new curve we get by tranformimg $K$ like this.
What is the equation of $E$?
The transformation is a dilation perpendicular to the given line. There are several ways to come up with a formula for it. For example, you could try rotating the coordinate system so that the line becomes the $x'$-axis. The dilation is then just a matter of multiplying the $y'$-coordinate by $3$, after which you rotate back the other way. Once you have the transformation formula, invert it and substitute into the circle equation.
You will find that the result is an ellipse. Indeed, the image of a circle under any affine transformation is some sort of ellipse. Knowing this, you can take a somewhat simpler approach. The circle is being stretched in a direction perpendicular to the given line. From this you can deduce both the principal axis directions and semiaxis lengths of the resulting ellipse. Since it’s not centered at the origin, its center will also move, so work out where the transformation moves this point and you should be able to derive an equation for the resulting ellipse without too much more work.