If a point is on a circle, its coordinates must satisfy the circle’s equation. What happens if you substitute $x = 4$ and $y = 5$ into each of the four equations?

For any circle that survives the point test, think about tangency: the distance from the center of that circle to the line $y = x + 1$ must equal the circle’s radius. How can you compute point-to-line distance using the formula for distance from a point to a line?

Once you can compute the distance from a center $(h,0)$ to the line $y = x + 1$, compare it to the radius implied by that choice’s equation. Which one matches exactly?

In Desmos, enter the line $y = x + 1$. Then type in each of the four answer choice equations as separate graphs so you can see all candidate circles at once.

Add the point $(4,5)$ in Desmos (you can type (4,5) directly). Look carefully to see which of the four circles goes exactly through that point; any circle that does not pass through the point can be eliminated.

For the circle(s) that pass through $(4,5)$, zoom in where the line and that circle meet. A tangent circle will touch the line at exactly one point, with the line just grazing the circle instead of cutting through it. Identify which equation corresponds to the circle that passes through $(4,5)$ and is tangent to the line.

The general form of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h,k)$ is the center and $r$ is the radius.

For the choices given, $k = 0$ in every case (because there is just $y^2$), so the centers and radii are:

• $(x + 9)^2 + y^2 = 50$: center $(-9,0)$, $r^2 = 50$
• $(x - 9)^2 + y^2 = 32$: center $(9,0)$, $r^2 = 32$
• $(x - 5)^2 + y^2 = 50$: center $(5,0)$, $r^2 = 50$
• $(x - 9)^2 + y^2 = 50$: center $(9,0)$, $r^2 = 50$

All four circles satisfy the first condition: their centers lie on the $x$-axis.

A point $(x,y)$ lies on a circle if its coordinates satisfy the circle’s equation.

For a circle with center $(h,0)$ and radius squared $r^2$, the point $(4,5)$ will be on the circle if

You can check this condition for each answer choice by plugging $x = 4$ and $y = 5$ into its equation and seeing whether the left-hand side equals the right-hand side.

A circle is tangent to a line if the distance from the center of the circle to the line is exactly equal to the radius.

The line is $y = x + 1$, which can be written as

For a circle with center $(h,0)$, the distance $d$ from $(h,0)$ to this line is

For a correct choice, this distance must equal the radius $r$ of that circle.

Now test the point $(4,5)$ in each equation:

• $(x + 9)^2 + y^2 = 50$:
  • Plug in $(4,5)$: $(4 + 9)^2 + 5^2 = 13^2 + 25 = 169 + 25 = 194 \ne 50$ → fails.
• $(x - 9)^2 + y^2 = 32$:
  • Plug in $(4,5)$: $(4 - 9)^2 + 5^2 = (-5)^2 + 25 = 25 + 25 = 50 \ne 32$ → fails.
• $(x - 5)^2 + y^2 = 50$:
  • Plug in $(4,5)$: $(4 - 5)^2 + 5^2 = (-1)^2 + 25 = 1 + 25 = 26 \ne 50$ → fails.
• $(x - 9)^2 + y^2 = 50$:
  • Plug in $(4,5)$: $(4 - 9)^2 + 5^2 = (-5)^2 + 25 = 25 + 25 = 50$ → works.

So only the circle $(x - 9)^2 + y^2 = 50$ passes through $(4,5)$. Finally, for this circle the center is $(9,0)$ and $r^2 = 50$, so $r = \sqrt{50} = \dfrac{10}{\sqrt{2}}$.

The distance from $(9,0)$ to the line $y = x + 1$ is

exactly equal to the radius, so it is tangent to the line. Therefore, the correct equation is $(x - 9)^2 + y^2 = 50$.