Let's solve the problem step by step.

Step 1: Equation of the line passing through (8,1) and (10,0)

To find the equation of the line, we use the slope-intercept form, $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

Finding the slope ($m$):

The slope $m$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 1}{10 - 8} = \frac{-1}{2}$ So, $m = -\frac{1}{2}$.

Using the point-slope form to find the equation:

Using the point (8,1): $y - y_1 = m(x - x_1)$ $y - 1 = -\frac{1}{2}(x - 8)$ $y - 1 = -\frac{1}{2}x + 4$ $y = -\frac{1}{2}x + 5$ So, the equation of the line is: $y = -\frac{1}{2}x + \frac{9}{2}$

Step 2: Equation of the circle centered at the origin and passing through (8,1)

The equation of a circle centered at the origin $(0, 0)$ with radius $r$ is: $x^2 + y^2 = r^2$ Since the circle passes through the point $(8, 1)$, we substitute these coordinates into the equation to find $r^2$: $8^2 + 1^2 = r^2$ $64 + 1 = r^2$ $r^2 = 65$ So, the equation of the circle is: $x^2 + y^2 = 65$

Step 3: Find the other intersection point of the line and the circle

To find the other point of intersection, we substitute the equation of the line $y = -\frac{1}{2}x + \frac{9}{2}$ into the equation of the circle $x^2 + y^2 = 65$.

Substitute $y = -\frac{1}{2}x + \frac{9}{2}$ into $x^2 + y^2 = 65$: $x^2 + \left(-\frac{1}{2}x + \frac{9}{2}\right)^2 = 65$ Expanding the square: $x^2 + \left(\frac{1}{4}x^2 - \frac{9}{2}x + \frac{81}{4}\right) = 65$ Simplifying: $x^2 + \frac{1}{4}x^2 - \frac{9}{2}x + \frac{81}{4} = 65$ Multiply through by 4 to eliminate the fraction: $4x^2 + x^2 - 18x + 81 = 260$ Simplify: $5x^2 - 18x - 179 = 0$ Now, solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where $a = 5$, $b = -18$, and $c = -179$: $x = \frac{18 \pm \sqrt{(-18)^2 - 4(5)(-179)}}{10}$ $x = \frac{18 \pm \sqrt{324 + 3580}}{10}$ $x = \frac{18 \pm \sqrt{3904}}{10}$ $x = \frac{18 \pm 62.48}{10}$ This gives two values for $x$: $x_1 = \frac{80.48}{10} = 8.048, \quad x_2 = \frac{-44.48}{10} = -4.448$ Since $(8,1)$ is already known, the other $x$-coordinate is $x_2 = -4.448$.

Step 4: Find the corresponding $y$-coordinate

Substitute $x = -4.448$ back into the equation of the line: $y = -\frac{1}{2}(-4.448) + \frac{9}{2}$ $y = 2.224 + 4.5 = 6.724$

So, the coordinates of the other point where the line and the circle meet are approximately: $(-4.448, 6.724)$

Would you like more details or have any questions?

Here are five related questions:

1. How do you derive the general form of a line equation from two points?
2. What is the significance of the slope in the line equation?
3. How do you find the radius of a circle given its equation?
4. How does substituting the line equation into the circle equation help find intersection points?
5. Can you solve the quadratic equation using a different method like completing the square?

Tip: When solving for intersection points between curves, always ensure to simplify your algebraic expressions step by step to avoid mistakes.