91Ó°ÊÓ

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows us to study geometry using a coordinate system. It is a powerful tool that connects algebra and geometry through graphs and equations.
The fundamental idea of coordinate geometry is to represent points, lines, and shapes using coordinates. In a 2D plane, we use the Cartesian coordinate system with two perpendicular axes - the x-axis and the y-axis.
In this system, every point is represented by an ordered pair \((x, y)\), where 'x' represents the horizontal distance from the origin (0,0) and 'y' represents the vertical distance. Coordinate geometry helps us to solve complex problems related to lines, curves, and shapes by converting them into algebraic equations.

To understand circles in coordinate geometry, we use the standard circle equation: \((x-h)^{2}+(y-k)^{2}=r^{2}\). This equation defines a circle with center at \( (h, k) \) and radius \((r)\).
For example, the circle equation in our exercise is \((x+1)^{2}+(y-3)^{2}=25\). Here, we can rewrite it in the form of \((x-(-1))^{2}+(y-3)^{2}=5^{2}\).
From this, we observe that:

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