If a circle passes through the points $(2,3)$ and $(4,5)$ and its center lies on the straight line $y - 4x + 3 = 0$, then its equation is......

Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$.

The center of the circle is $(-g, -f)$.

Since the center lies on the line $y - 4x + 3 = 0$:

$-f - 4(-g) + 3 = 0$

$4g - f + 3 = 0$

The circle passes through $(2, 3)$, so substituting $x = 2, y = 3$:

$2^2 + 3^2 + 2g(2) + 2f(3) + c = 0$

$4g + 6f + c + 13 = 0$

The circle passes through $(4, 5)$, so substituting $x = 4, y = 5$:

$4^2 + 5^2 + 2g(4) + 2f(5) + c = 0$

$8g + 10f + c + 41 = 0$

Subtracting the equation for $(2, 3)$ from the equation for $(4, 5)$:

$(8g - 4g) + (10f - 6f) + (c - c) + (41 - 13) = 0$

$4g + 4f + 28 = 0$

$g + f + 7 = 0$

From the condition $4g - f + 3 = 0$, we have $f = 4g + 3$. Substituting this value:

$g + (4g + 3) + 7 = 0$

$5g + 10 = 0$

$g = -2$

Substituting the value of $g$ to find $f$:

$f = 4(-2) + 3 = -5$

Substituting the values of $g$ and $f$ to find $c$:

$4(-2) + 6(-5) + c + 13 = 0$

$-8 - 30 + c + 13 = 0$

$c - 25 = 0$

$c = 25$

The equation of the circle is:

$x^2 + y^2 + 2(-2)x + 2(-5)y + 25 = 0$

$x^2 + y^2 - 4x - 10y + 25 = 0$