Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, – 3) and B is (1, 4).
Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, – 3) and B is (1, 4).
Solution and Explanation
Let the coordinates of point A be (x, y).
Mid-point of AB is (2, −3), which is the center of the circle.
\(\therefore\,(2,-3)=(\frac{x+1}{2},\frac{y+4}{2})\)
\(\Rightarrow \, \frac{x+1}{2}=2\, and\,\frac{y+4}{2}=-3\)
\(\Rightarrow \,x+1=4\,and\,y+4=-6\)
\(\Rightarrow \,x=3\,and\,y=-10\)
Therefore, the coordinates of A are (3,-10)
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The central idea in coordinate geometry is to assign numerical coordinates to points in a plane or space, which allows us to describe their positions and relationships using algebraic equations. The most common coordinate system is the Cartesian coordinate system, named after the French mathematician and philosopher René Descartes.