Question

find the standard equation of the circle passing through a given point with a given center. center (4,1) and passing through (3,5)

Explanation:

Step1: Recall the standard - form of a circle equation

The standard - form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. Here, the center $(h,k)=(4,1)$, so the equation of the circle is $(x - 4)^2+(y - 1)^2=r^2$.

Step2: Calculate the radius

The circle passes through the point $(3,5)$. Substitute $x = 3$ and $y = 5$ into the equation $(x - 4)^2+(y - 1)^2=r^2$.
\[

$$\begin{align*} (3 - 4)^2+(5 - 1)^2&=r^2\\ (-1)^2+4^2&=r^2\\ 1 + 16&=r^2\\ r^2&=17 \end{align*}$$

\]

Step3: Write the final equation

Substitute $r^2 = 17$ into the equation $(x - 4)^2+(y - 1)^2=r^2$. The standard equation of the circle is $(x - 4)^2+(y - 1)^2=17$.

Answer:

$(x - 4)^2+(y - 1)^2=17$