What is the equation of the circle passing through ( 12 , 1 ) (12,1) and ( 2 , − 3 ) (2,−3) with center on the line 2 x − 5 y + 10

To find the equation of the circle that passes through the points $(12, 1)$ and $(2, -3)$ with its center on the line $2x - 5y + 10 = 0$, we can follow these steps:

Step 1: General Equation of the Circle

The general equation of a circle can be expressed as:

$(x - h)^2 + (y - k)^2 = r^2$

where $(h, k)$ is the center and $r$ is the radius.

Step 2: Center on the Given Line

The center $(h, k)$ must satisfy the line equation $2h - 5k + 10 = 0$. We can express $k$ in terms of $h$:

$k = \frac{2h + 10}{5}$

Step 3: Distance from the Center to the Points

The circle passes through the points $(12, 1)$ and $(2, -3)$. Therefore, the distances from the center to these points should be equal to the radius $r$:

1. From $(h, k)$ to $(12, 1)$: $r^2 = (h - 12)^2 + (k - 1)^2$

2. From $(h, k)$ to $(2, -3)$: $r^2 = (h - 2)^2 + (k + 3)^2$

Step 4: Setting the Equations Equal

Setting the two equations for $r^2$ equal gives:

$(h - 12)^2 + (k - 1)^2 = (h - 2)^2 + (k + 3)^2$

Expanding both sides:

$(h^2 - 24h + 144) + (k^2 - 2k + 1) = (h^2 - 4h + 4) + (k^2 + 6k + 9)$

Step 5: Simplifying the Equation

Cancelling $h^2$ and $k^2$ from both sides results in:

$-24h + 144 - 2k + 1 = -4h + 4 + 6k + 9$

Simplifying further:

$-20h + 132 - 8k = 0$

Rearranging gives:

$20h + 8k = 132 \quad \Rightarrow \quad 5h + 2k = 33$

Step 6: Substitute for $k$

Now substitute $k = \frac{2h + 10}{5}$ into the equation $5h + 2k = 33$:

$5h + 2\left(\frac{2h + 10}{5}\right) = 33$

Multiply through by 5 to eliminate the fraction:

$25h + 2(2h + 10) = 165$

Expanding gives:

$25h + 4h + 20 = 165 \quad \Rightarrow \quad 29h = 145 \quad \Rightarrow \quad h = 5$

Step 7: Find $k$

Now substitute $h = 5$ back to find $k$:

$k = \frac{2(5) + 10}{5} = \frac{20}{5} = 4$

Step 8: Find the Radius

Now we can find $r^2$ using either point. Using point $(12, 1)$:

$r^2 = (5 - 12)^2 + (4 - 1)^2 = (-7)^2 + (3)^2 = 49 + 9 = 58$

Step 9: Final Equation of the Circle

The equation of the circle is:

$(x - 5)^2 + (y - 4)^2 = 58$

This can be expanded to:

$x^2 - 10x + 25 + y^2 - 8y + 16 = 58$

Simplifying gives:

$x^2 + y^2 - 10x - 8y - 17 = 0$

Final Result

Thus, the equation of the circle is:

$x^2 + y^2 - 10x - 8y - 17 = 0$

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Would you like further details or explanations on any part of this process? Here are some related questions you might find interesting:

1. How do you derive the radius from the center and a point on the circle?
2. What methods can be used to find the center of a circle given points?
3. How do you convert the standard form of a circle into general form?
4. Can you explain the geometric interpretation of a circle's center and radius?
5. What are the characteristics of circles that intersect two specific points?

Tip: Always check your calculations by substituting the points back into the final equation to ensure they satisfy it.