Do You Know Your Quadrants?

First, we need to find the intersection of the circle and the x and y axes. This can be done more easily by converting the equation of the circle to its standard form.

$({ x }^{ 2 }\quad +\quad { y }^{ 2 }\quad +\quad 10x\quad -\quad 14y\quad -\quad 95\quad =\quad 0\\ \Rightarrow \quad ({ x }^{ 2 }\quad +\quad 10x\quad +\quad { { 5 }^{ 2 })\quad +\quad (y }^{ 2 }\quad -\quad 14y\quad +\quad { 7 }^{ 2 })\quad =\quad 95\quad +\quad { 5 }^{ 2 }\quad +\quad { 7 }^{ 2 }\\ \Rightarrow \quad { (x+5) }^{ 2 }\quad +\quad { (y-7) }^{ 2 }\quad =\quad 169$

Now we can find the x and y intercepts by setting $y$ to 0 and $x$ to 0 respectively.

${ (x+5) }^{ 2 }\quad +\quad { (0-7) }^{ 2 }\quad =\quad 169\\ \Rightarrow \quad { (x+5) }^{ 2 }\quad =\quad 120\\ \Rightarrow \quad x\quad =\quad -5\quad \pm \quad \sqrt { 120 } \\ { (0+5) }^{ 2 }\quad +\quad { (y-7) }^{ 2 }\quad =\quad 169\\ \Rightarrow \quad { (y-7) }^{ 2 }\quad =\quad 144\\ \Rightarrow \quad y\quad =\quad 7\quad \pm \quad 12$

The $x$ values are positive and the $y$ values are negative in quadrant $IV$ , so we're left with $(0,\quad -5)$ and $(\sqrt { 120 } \quad -\quad 5,\quad 0)$ as our intersection points. Now, to find the amount of the circle's circumference in the fourth quadrant, we simply need to find the angle between the lines formed by the intersection points and the center of the circle at $(-5,\quad 7)$ . To do this, we first need to find the slopes of such lines.

For the line connecting the center of the circle to the y-intercept at $(0,\quad -5)$ , we have a slope of ${ m }_{ 1 }\quad =\quad -\frac { 12 }{ 5 }$ . For the other line, we have a slope of ${ m }_{ 2 }\quad =\quad -\frac { 7 }{ \sqrt { 120 } }$

Finally, we now have the slopes, so we can now find the angle between the lines.

$\tan { (\theta ) } \quad =\quad \frac { { m }_{ 2 }\quad -\quad { m }_{ 1 } }{ 1\quad +\quad { m }_{ 2 }{ m }_{ 1 } } \quad =\quad \frac { 7\sqrt { 30 } \quad -\quad 30 }{ 12 } \\ \Rightarrow \quad \theta \quad \approx \quad 34.801°$

The entire circle has an angle of 360 degrees, so just simply divide theta by 360 to get the percentage, so the answer is $\approx \boxed { 9.667 }$ percent, correct up to 3 decimal places.