Find the position of the point (3,-4) with respect to the circle `x^(2)+y^(2)=36`.

Answer

To determine the position of the point (3, -4) with respect to the circle defined by the equation \(x^2 + y^2 = 36\), we can follow these steps:

### Step 1: Identify the center and radius of the circle
The equation of the circle is given as \(x^2 + y^2 = 36\). This can be compared to the standard form of a circle's equation, which is \(x^2 + y^2 = r^2\), where \(r\) is the radius.

- From \(x^2 + y^2 = 36\), we can see that:
- The center of the circle is at the origin (0, 0).
- The radius \(r\) is given by \(r = \sqrt{36} = 6\).

### Step 2: Calculate the distance from the point (3, -4) to the center of the circle (0, 0)
To find the position of the point (3, -4) with respect to the circle, we need to calculate the distance from the point to the center of the circle using the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, \((x_1, y_1) = (0, 0)\) (the center of the circle) and \((x_2, y_2) = (3, -4)\) (the point in question).

Substituting the values into the formula:

\[ d = \sqrt{(3 - 0)^2 + (-4 - 0)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Compare the distance with the radius
Now that we have the distance \(d = 5\) units, we compare this with the radius of the circle, which is \(6\) units.

- Since \(5 < 6\), we conclude that the point (3, -4) lies inside the circle.

### Final Conclusion
The position of the point (3, -4) with respect to the circle \(x^2 + y^2 = 36\) is that it lies **inside** the circle.

---