Question

The curvature of the circle \(x^{2}+y^{2}=25\) at the point \((3,4)\) is

Short answer

The curvature at the point \((3,4)\) is \(\frac{1}{5}\).

Step-by-step solution

Write the equation of the circle

The equation provided for the circle is \(x^{2} + y^{2} = 25\). This is a standard form of a circle's equation where the circle is centered at \((0,0)\) and has a radius of \(r^2 = 25\), therefore, \(r = 5\).

Recall the formula for curvature

For a circle, the curvature \(\kappa\) is given by the formula \(\kappa = \frac{1}{r}\), where \(r\) is the radius of the circle.

Calculate the curvature

Using the formula \(\kappa = \frac{1}{r}\) and the radius \(r = 5\), substitute into the formula: \(\kappa = \frac{1}{5}\).

Verify the point is on the circle

Plug the coordinates \((3,4)\) into the equation \(x^{2} + y^{2} = 25\) to verify: \(3^2 + 4^2 = 9 + 16 = 25\). The point lies on the circle, and thus the curvature is valid at this point.

Key concepts

Equation of a Circle

To understand a circle's properties, recognizing its equation is fundamental. The equation of a circle in a standard form is \(x^2 + y^2 = r^2\). This represents a circle centred at the origin \(0,0\) with radius \(r\).
In the given exercise, the circle's equation is \(x^2 + y^2 = 25\). From this equation, we can deduce that\(r^2 = 25\), giving us \(r = 5\). This tells us the circle is centered at the origin and stretches out 5 units in every direction from \(0,0\).
Understanding this representation helps in visualizing and solving problems related to circles, like finding curvature or verifying if a point lies on the circle.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. It is a constant measurement and crucial in various calculations involving the circle.
Given the equation \(x^2 + y^2 = 25\), we identified earlier that \(r = 5\). This length defines the size of our circle. The radius is a critical component not only in determining the curvature but also in other calculations involving area and circumference.
For any practical applications involving circles, knowing the radius allows one to better comprehend the circle's dimensions and characteristics.

Curvature Formula

Curvature provides insight into how sharply a circle curves at any given point. For a circle, this measure is fortunately straightforward due to its uniform curvature at any point along its circumference.
The curvature \(\kappa\) of a circle is given by the formula \(\kappa = \frac{1}{r}\).
This equation demonstrates that the curvature is inversely proportional to the radius: a smaller radius brings a bigger curvature (sharper turning), and a larger radius signifies a smaller curvature (flatter appearance).
In our problem, substituting \(r = 5\) results in \(\kappa = \frac{1}{5}\). This simple calculation provides clear insights into the circle's bending nature at any point on its circumference.

Verification of a Point on a Circle

Verifying whether a point lies on a circle is straightforward if you have the circle's equation. The point satisfies the circle's equation if, upon substitution into \(x^2 + y^2 = r^2\), the equation holds true.
With our circle's equation \(x^2 + y^2 = 25\), verifying the point \(3,4\) involves substituting these coordinates into the equation. Calculating, \(3^2 + 4^2 = 9 + 16 = 25\), confirms that the point does indeed reside on the circle.
This verification process is essential for ensuring accuracy in calculations related to curvature, tangents, and other circle-related properties at specific points.