The question asks for the position of the point $P(5, 12)$ with respect to the circle given by the equation:

$x^2 + y^2 = 16$

Steps to solve:

1. Equation of the Circle: The given equation, $x^2 + y^2 = 16$, represents a circle with its center at $(0, 0)$ and a radius $r = \sqrt{16} = 4$.

2. Distance from the Center to Point $P(5, 12)$: The distance $d$ between the center of the circle $(0, 0)$ and the point $P(5, 12)$ can be found using the distance formula: $d = \sqrt{(5 - 0)^2 + (12 - 0)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

3. Compare the Distance to the Radius:

   • The radius of the circle is 4.
   • The distance from the center to $P(5, 12)$ is 13.

Since the distance $d = 13$ is greater than the radius $r = 4$, the point $P(5, 12)$ lies outside the circle.

Conclusion:

The position of the point $P(5, 12)$ with respect to the circle is outside.

Would you like further details or have any questions?

Here are 5 related questions you might explore:

1. What is the equation of a circle with center $(h, k)$ and radius $r$?
2. How do you find the distance between two points in a plane?
3. How can you check if a point lies inside, on, or outside a circle?
4. What is the general form of the equation of a circle in the coordinate plane?
5. How does the distance formula relate to the Pythagorean Theorem?

Tip: For any point $P(x_1, y_1)$, compare the distance from the center to $P$ with the circle's radius to determine its relative position.