In the given figure, we need to find the equations for both the circle and the line.

1. Equation of the Circle:

The circle is centered at the origin, $(0, 0)$, and passes through the point $(-5, 12)$.

The general equation of a circle with center $(h, k)$ and radius $r$ is: $(x - h)^2 + (y - k)^2 = r^2$ Since the center is at $(0, 0)$, the equation simplifies to: $x^2 + y^2 = r^2$

We can determine $r$ by finding the distance from the center $(0, 0)$ to the point $(-5, 12)$, which lies on the circle: $r = \sqrt{(-5 - 0)^2 + (12 - 0)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

Thus, the equation of the circle is: $x^2 + y^2 = 13^2 = 169$ $x^2 + y^2 = 169$

2. Equation of the Line:

The line is passing through the point $(-5, 12)$ and is perpendicular to the radius of the circle at that point. The slope of the radius from the center $(0, 0)$ to $(-5, 12)$ is: $\text{slope of radius} = \frac{12 - 0}{-5 - 0} = -\frac{12}{5}$

Since the line is perpendicular to the radius, its slope will be the negative reciprocal of $-\frac{12}{5}$, which is: $\text{slope of line} = \frac{5}{12}$

Now, using the point-slope form of a line equation: $y - y_1 = m(x - x_1)$ Substituting $(-5, 12)$ and the slope $\frac{5}{12}$: $y - 12 = \frac{5}{12}(x + 5)$

Expanding this equation: $y - 12 = \frac{5}{12}x + \frac{25}{12}$ Multiplying the entire equation by 12 to eliminate the fraction: $12y - 144 = 5x + 25$ Rearranging: $5x - 12y = -169$

Thus, the equation of the line is: $5x - 12y = -169$

Summary:

• The equation of the circle is: $x^2 + y^2 = 169$
• The equation of the line is: $5x - 12y = -169$

Would you like more details or have any other questions?

Here are some related questions:

1. How do you determine the radius of a circle given its center and a point on the circle?
2. What is the significance of the negative reciprocal in the slope of perpendicular lines?
3. How do you derive the general equation of a line from the point-slope form?
4. What other geometric properties can be derived from a circle and a tangent line?
5. How can you verify if a point lies on both the circle and the line?

Tip: Always check if a line is tangent to a circle by verifying that the distance between the center of the circle and the line equals the radius of the circle.