Question

The radius of a circle with centre \((-2,3)\) is 5 units. The point \((2,5)\) lies (1) on the circle (2) inside the circle (3) outside the circle (4) None of these

Short answer

Known information: - Center of the circle: (-2, 3) - Radius: 5 units - Point: (2, 5) Answer: The point (2, 5) lies inside the circle.

Step-by-step solution

Write down known information

We are given the coordinates of the centre of the circle, \((-2, 3)\), and the radius, 5 units. The point we need to analyze is \((2, 5)\). Our task is to find out the relationship between the given point and the circle.

Calculate the distance between the center of the circle and the given point

We can calculate the distance between the points using the distance formula: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\). In this case, let's plug the coordinates of the center \((-2, 3)\) and the point \((2, 5)\) into the formula: \(d=\sqrt{(-2-2)^2+(3-5)^2} =\sqrt{(-4)^2+(-2)^2}=\sqrt{16+4}=\sqrt{20}\) So the distance between the center of the circle and the given point is \(\sqrt{20}\).

Compare the distance to the radius to determine the position

We now compare the distance \(\sqrt{20}\) to the radius, 5 units: - If the distance is equal to the radius, the point is on 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. Since \(\sqrt{20} \approx 4.47 < 5\), the distance is smaller than the radius.

Conclude

Since the distance between the center of the circle and the point is less than the radius, we can conclude that the point \((2,5)\) lies inside the circle (option 2).

Key concepts

Distance Formula

The distance formula is a crucial tool in geometry that helps us determine the distance between two points on a coordinate plane. Whenever you're given two points with coordinates \(x_1, y_1\) and \(x_2, y_2\), you can calculate the distance between these two points using:\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.\]This formula essentially applies the Pythagorean theorem to find the straight line distance between the points. It works by subtracting the coordinates of the points to form a right triangle, allowing you to measure the hypotenuse.

• Always subtract coordinate values carefully to ensure accuracy.
• Simplify your results properly. In many cases, you might need to approximate square roots.

Learning to apply the distance formula effectively will greatly aid in solving various geometry problems, especially in contexts like determining a point’s relation to a circle.

Circle Equation

A circle's equation on a coordinate plane is a way to mathematically describe all its points using the circle's center and radius. The standard equation for a circle with center at \(h, k\) and radius \(r\) is:\[(x - h)^2 + (y - k)^2 = r^2\]The circle equation tells us that the distance from any point on the circle to its center is always equal to the radius.

• The \(h\) and \(k\) are simply the coordinates of the circle's center.
• It's important to square the radius on the right side of the equation as shown.

The circle equation is foundational because it not only helps identify if a point is on the circle but, by manipulation, can show whether a point lies inside or outside of it. Practicing with this equation boosts your confidence in dealing with circle-related problems.

Coordinates of Points

Coordinates are a way to see where points are located on a plane. Every point in a two-dimensional space has an \(x\) and \(y\) coordinate written as \(x,y\).

• The \(x\) coordinate determines the horizontal position.
• The \(y\) coordinate determines the vertical position.

This coordinate system is fundamental in geometry, making it easy to calculate distances, slopes, and understand shapes. For example, in our problem, determining the circle’s center at \((-2, 3)\) and evaluating the point \((2, 5)\) helped us understand where the point lies in relation to the circle.Remember:- Positive \(x\) means right of the origin.- Positive \(y\) means above the origin.Understanding coordinates allows you to navigate through various mathematical concepts seamlessly.