The given problem describes a curve with the equation:

$x = y^2 - 6y + 11$

This curve is rotated 90° clockwise about a point $P$, which has coordinates $(-2, 3)$, to obtain a new curve $C$. The task is to find the equation of the new curve $C$.

Step 1: Equation of the Original Curve

The original curve is: $x = y^2 - 6y + 11$ This can also be written in the form $f(y)$, meaning it relates $x$ as a function of $y$.

Step 2: Shift the Coordinates to the Origin

Since we are rotating around the point $P(-2, 3)$, we first need to shift the coordinates so that $P$ becomes the origin. To do this, we perform the following translation:

• Let $x' = x + 2$
• Let $y' = y - 3$

Thus, the original curve becomes: $x' - 2 = (y' + 3)^2 - 6(y' + 3) + 11$ Simplifying the right-hand side:

$(x' - 2) = (y'^2 + 6y' + 9) - 6(y' + 3) + 11$ $x' - 2 = y'^2 + 6y' + 9 - 6y' - 18 + 11$ $x' - 2 = y'^2 + 2$ $x' = y'^2 + 4$

So the equation in the shifted coordinates is: $x' = y'^2 + 4$

Step 3: Rotate the Curve 90° Clockwise

To rotate the curve 90° clockwise around the origin, we apply the following transformation: $x'' = y'$ $y'' = -x'$

Thus, the new curve in the rotated coordinates is: $y'' = -(y'^2 + 4)$ or equivalently: $y'' = -y'^2 - 4$

Step 4: Shift Back to the Original Coordinates

Now, we reverse the shift to return to the original coordinates $x$ and $y$: $x = x'' + 2$ $y = y'' + 3$

Substitute $y'' = -y'^2 - 4$ into the second equation: $y = -y'^2 - 4 + 3$ $y = -y'^2 - 1$

Thus, the final equation of the new curve $C$ is: $y = -(x - 2)^2 - 1$

This is the equation of the curve $C$ after a 90° clockwise rotation about the point $P(-2, 3)$.

Would you like further clarification or details?

Here are 5 related questions for practice:

1. How would the equation change if the rotation was 90° counterclockwise?
2. What is the general transformation matrix for rotating a point around an arbitrary point?
3. How would you solve for the new equation if the curve was translated instead of rotated?
4. What are the key differences in the equations of curves when rotating by different angles (e.g., 180°, 270°)?
5. How do you find the points of intersection between two rotated curves?

Tip: Always break down rotations into translations to the origin and apply coordinate transformations step by step to avoid mistakes.