Answer:
circle: x² + y² = 169
line: y = (-12/5)x
Step-by-step explanation:
First, we can solve for the radius of the circle by using the distance formula:
[tex]\displaystyle d=\sqrt{(d_x)^2 + (d_y)^2}{}[/tex]
where:
- [tex]d_x[/tex] is the distance covered parallel to the x-axis
- [tex]d_y[/tex] is the distance covered parallel to the y-axis
- [tex]d[/tex] is the absolute distance covered (like the hypotenuse of a right triangle)
↓↓↓ plugging in the given values
[tex]r = \sqrt{5^2 + 12^2}[/tex]
[tex]r = \sqrt{25+144}[/tex]
[tex]r=\sqrt{169}[/tex]
[tex]r=13[/tex]
Now, using this value, we can construct an equation for the circle, because we also know that its center is:
Using the equation for a circle:
[tex](x-h)^2+(y-k)^2 = r^2[/tex]
↓ plugging in the known values
[tex](x-0)^2+(y-0)^2 = 13^2[/tex]
[tex]\boxed{x^2+y^2=169}[/tex]
Next, we can find the equation of the line using slope-intercept form:
y = mx + b
where:
We can see that:
- m = rise / run = -12 / 5
- b = 0
So, the slope-intercept form equation for the line is:
y = (-12/5)x + 0
which simplifies to:
y = (-12/5)x
