A circle has its center on straight line 5x-2y+1=0 and cuts the x axis at two points whose absciasse are -5,3 find equation of the circle and it's radius.

To find the equation of the circle that meets the given conditions, we need to break down the problem step by step. The circle has its center on the line defined by the equation \(5x - 2y + 1 = 0\) and intersects the x-axis at the points where the x-coordinates (abscissas) are -5 and 3. Let's go through the process together.

Identifying Key Points

First, we note the points where the circle intersects the x-axis. These points are:

• A(-5, 0)
• B(3, 0)

Finding the Center of the Circle

The center of the circle, which we will denote as \(C(h, k)\), lies on the line \(5x - 2y + 1 = 0\). To find the center, we need to determine its coordinates. The center must also be equidistant from both points A and B since it is the midpoint of the diameter of the circle.

Calculating the Midpoint

The midpoint \(M\) of segment AB can be calculated using the midpoint formula:

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{-5 + 3}{2}, \frac{0 + 0}{2}\right) = \left(-1, 0\right)

Finding the Radius

The radius \(r\) of the circle is the distance from the center \(C(h, k)\) to either point A or B. Since we have the coordinates of point A, we can express the radius as:

r = distance from C to A = \sqrt{(h + 5)^2 + (k - 0)^2}

Substituting the Center into the Line Equation

Now, since the center lies on the line \(5x - 2y + 1 = 0\), we can substitute \(h\) and \(k\) into this equation:

5h - 2k + 1 = 0

Using the Midpoint to Find the Center

We know the midpoint is \((-1, 0)\), so we can set \(h = -1\) and \(k = 0\). Substituting these values into the line equation:

5(-1) - 2(0) + 1 = -5 + 1 = -4 \neq 0

This means the center cannot be at the midpoint. Instead, we need to find a point on the line that is equidistant from both points A and B.

Finding the Correct Center

Let’s express the center as \(C(h, k)\) where \(k\) can be expressed in terms of \(h\) using the line equation:

2k = 5h + 1 → k = \frac{5h + 1}{2}

Setting Up the Distance Equation

Now, we can set the distance from the center to point A equal to the distance from the center to point B:

\(\sqrt{(h + 5)^2 + \left(k - 0\right)^2} = \sqrt{(h - 3)^2 + \left(k - 0\right)^2}\)

Squaring both sides eliminates the square roots:

(h + 5)^2 = (h - 3)^2

Expanding and Solving

Expanding both sides:

h^2 + 10h + 25 = h^2 - 6h + 9

Combining like terms gives:

16h + 16 = 0 → h = -1

Finding k

Now substituting \(h = -1\) back into the equation for \(k\):

k = \frac{5(-1) + 1}{2} = \frac{-5 + 1}{2} = \frac{-4}{2} = -2

Thus, the center of the circle is at \(C(-1, -2)\).

Calculating the Radius Again

Now we can find the radius using the distance from the center to point A:

r = \sqrt{(-1 + 5)^2 + (-2 - 0)^2} = \sqrt{(4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}

Final Equation of the Circle

The standard form of the equation of a circle is given by:

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

Substituting the values of \(h\), \(k\), and \(r\):

(x + 1)^2 + (y + 2)^2 = (2\sqrt{5})^2

(x + 1)^2 + (y + 2)^2 = 20

In summary, the equation of the circle is:

(x + 1)^2 + (y + 2)^2 = 20

And the radius of the circle is \(2\sqrt{5}\).