Problem 66 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 65-66: Determine whether the point \(P\) is inside, outside, or on the circle with center \(C\) and radius \(r\). (a) \(P(3,8), \quad C(-2,-4), \quad r=13\) (b) \(P(-2,5), \quad C(3,7), \quad r=6\) (c) \(P(1,-2), \quad C(6,-7), \quad r=7\)

Step-by-Step Solution

Answer

(a) On the circle, (b) Outside the circle, (c) Inside the circle.

Step 1: Understand the Condition to Check

For each part, compare the distance between the point \(P(x, y)\) and the circle's center \(C(h, k)\) to the radius \(r\). If the distance is less than \(r\), \(P\) is inside the circle; if equal to \(r\), \(P\) is on the circle; if more, \(P\) is outside the circle.

Step 2: Use the Distance Formula

To find the distance between \(P(a, b)\) and \(C(h, k)\), use the formula: \( d = \sqrt{(a-h)^2 + (b-k)^2} \).

Step 3a: Calculate for Part (a)

Given \(P(3,8), C(-2,-4), r=13\). The distance is \(d = \sqrt{(3-(-2))^2 + (8-(-4))^2} = \sqrt{(3+2)^2 + (8+4)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\). \(P\) is on the circle since \(d = r\).

Step 3b: Calculate for Part (b)

Given \(P(-2,5), C(3,7), r=6\). The distance is \(d = \sqrt{(-2-3)^2 + (5-7)^2}= \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \). \(P\) is outside the circle since \(d > r\).

Step 3c: Calculate for Part (c)

Given \(P(1,-2), C(6,-7), r=7\). The distance is \(d = \sqrt{(1-6)^2 + (-2-(-7))^2} = \sqrt{(-5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} \). \(P\) is inside the circle since \(d < r\).

Key Concepts

Distance Formula

The distance formula is a crucial tool in geometry, particularly in analytic geometry. It is used to calculate the distance between two points on a coordinate plane. This formula is derived from the Pythagorean theorem. When you have two points, say point \(P(x_1, y_1)\) and point \(Q(x_2, y_2)\), you can determine the distance between these points using:

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

This calculation provides the direct line distance connecting the two points, which is often referred to as the "straight-line distance."
In circle geometry, this formula helps us understand the position of any point relative to the circle's center.

Circle Equation

The equation of a circle is foundational for analyzing points in relation to the circle. A general circle equation in a Cartesian plane with center at \(C(h, k)\) and radius \(r\) is expressed as:

• \[ (x - h)^2 + (y - k)^2 = r^2 \]

This equation states that any point \(P(x, y)\) lying on the circle will satisfy this equation, meaning its distance squared from the center \(C\) will equal the radius squared.
When assessing the position of a point with respect to the circle, we're essentially checking if this equation holds true, or if the left-hand value is less or greater than \(r^2\).

Point Position Relative to Circle

Determining a point's position relative to a circle uses both the circle equation and the distance formula. You compare the computed distance from the point to the circle's center with the circle's radius:

• If the distance equals the radius, the point lies right on the boundary of the circle.
• If the distance is less than the radius, the point is inside the circle.
• If the distance is greater than the radius, the point is outside the circle.

This methodology is applied by computing the distance with the distance formula and assessing against the radius, as exemplified in the exercise steps you reviewed earlier.

Analytic Geometry

Analytic geometry, also known as coordinate geometry, merges algebra and geometry using a coordinate system. It allows geometric problems to be expressed as algebraic equations.
In the context of circles in analytic geometry, you use numerical coordinates and equations to handle tasks like determining the position of a point relative to a circle.
This branch of mathematics is essential for solving many real-world problems where geometry meets algebra, such as in physics, engineering, and computer graphics. By utilizing structures like the distance formula and circle equation analytically, you can make precise calculations about geometric forms.