What is "if point (4?

Here's information about the concept of checking if a point (4, -5) lies inside the circle defined by the equation (x-2)² + (y+1)² = 13:

The question is about determining whether a specific point resides within a given <a href="https://www.wikiwhat.page/kavramlar/Circle" >Circle</a>. The circle is defined by its equation in standard form. The point is (4, -5).

The core method involves substitution and comparison.

1. Understanding the Circle Equation: The equation (x-2)² + (y+1)² = 13 represents a circle centered at (2, -1) with a radius squared (r²) of 13. Therefore, the radius (r) is √13.

2. Substitution: Substitute the x and y coordinates of the point (4, -5) into the left-hand side of the circle's equation:

   (4-2)² + (-5+1)²

3. Calculation: Simplify the expression:

   (2)² + (-4)² = 4 + 16 = 20

4. Comparison: Now, compare the calculated value (20) with the right-hand side of the circle's equation (13).

   • If the calculated value is less than 13, the point lies inside the circle.
   • If the calculated value is equal to 13, the point lies on the circle.
   • If the calculated value is greater than 13, the point lies outside the circle.

5. Conclusion: In this case, 20 > 13. Therefore, the point (4, -5) lies outside the circle.

In summary, checking if a point is inside a circle is an application of <a href="https://www.wikiwhat.page/kavramlar/Analytic%20Geometry" >Analytic Geometry</a> principles, specifically relating to the <a href="https://www.wikiwhat.page/kavramlar/Equation%20of%20a%20Circle" >Equation of a Circle</a>. The <a href="https://www.wikiwhat.page/kavramlar/Distance%20Formula" >Distance Formula</a> is implicitly used here as the equation of a circle is derived from calculating the distance between a point on the circle and its center.