Coordinate Representation Math Example 2

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Example 2

medium
Write the equation of the circle with centre (−2,3)(-2, 3) and radius 55 in standard form. Verify that the point (3,3)(3, 3) lies on the circle.

Solution

  1. 1
    Step 1: Standard form of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 with centre (h,k)(h, k) and radius rr.
  2. 2
    Step 2: (x−(−2))2+(y−3)2=25⇒(x+2)2+(y−3)2=25(x-(-2))^2 + (y-3)^2 = 25 \Rightarrow (x+2)^2 + (y-3)^2 = 25.
  3. 3
    Step 3: Check (3,3)(3, 3): (3+2)2+(3−3)2=25+0=25(3+2)^2 + (3-3)^2 = 25 + 0 = 25. ✓ The point lies on the circle.

Answer

(x+2)2+(y−3)2=25(x+2)^2 + (y-3)^2 = 25; yes, (3,3)(3,3) is on the circle.
Coordinate representation allows us to express geometric shapes as algebraic equations. The circle equation follows directly from the definition: all points at distance rr from the centre. Substituting a point verifies membership.

About Coordinate Representation

Describing geometric objects precisely using ordered pairs (x,y)(x, y) or triples (x,y,z)(x, y, z) in a coordinate system.

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More Coordinate Representation Examples

Example 1 easy

Plot points [formula], [formula], [formula] on the coordinate plane and find the distance from [form

Example 3 easy

Find the midpoint of the segment joining [formula] and [formula].

Example 4 hard

Three vertices of a parallelogram are [formula], [formula], [formula]. Using coordinate methods, fin

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