Question

find the equation for the circle with center (4,5) and passing through (3,4).
which is the correct equation?
a. $(x + 4)^2+(y + 5)^2 = 4$
b. $(x + 4)^2+(y + 5)^2 = 81$
c. $(x - 4)^2+(y - 5)^2 = 49$
d. $(x - 4)^2+(y - 5)^2 = 2$

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 and \(r\) is the radius.

Step2: Determine the center values

Given the center \((4,5)\), so \(h = 4\) and \(k = 5\). The equation becomes \((x - 4)^2+(y - 5)^2=r^2\).

Step3: Calculate the radius

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the radius. Here, \((x_1,y_1)=(4,5)\) (center) and \((x_2,y_2)=(3,4)\) (a point on the circle).

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Step4: Find \(r^2\)

Since \(r=\sqrt{2}\), then \(r^2 = (\sqrt{2})^2=2\).

Step5: Write the final equation

Substitute \(r^2 = 2\) into \((x - 4)^2+(y - 5)^2=r^2\), we get \((x - 4)^2+(y - 5)^2=2\).

Answer:

D. \((x - 4)^2+(y - 5)^2=2\)