find the center of a circle passing through the points (6, -6) (3, -7) and (3, 3) also find the radius
Answers
Answer:
- Radius = 5 units
- Center of circle = (6,-1)
Given,
- Circle formed by passing through three points: (6, -6) (3, -7) and (3, 3)
To find,
- Radius of the circle formed.
Solution:
We know that the distance between the center and the point is equal to the radius. Let us assume the center to be: (x,y)
→ Dist. b/w Center to Point 1 = Dist b/w Center to Point 2
→ ( x - 6 )² + ( y + 6 )² = ( x - 3 )² + ( y + 7 )²
→ ( x² -12x + 36 ) + ( y² + 12y + 36 ) = ( x² - 6x + 9 ) + ( y² + 14y + 49 )
→ ( x² - 12x + y² + 12y + 72 ) = ( x² - 6x + y² + 14y + 58 )
Cancelling the common terms we get:
→ 12y - 12x + 72 = 14y - 6x + 58
→ 12x - 6x + 14y - 12y = 72 - 58
→ 6x + 2y = 14
Simplifying the above equation we get:
→ 3x + y = 7 ...(i)
Now considering Point 1 and Point 3 we get:
→ ( x - 6 )² + ( y + 6 )² = ( x - 3 )² + ( y - 3 )²
→ ( x² -12x + 36 ) + ( y² + 12y + 36 ) = ( x² - 6x + 9 ) + ( y² - 6y + 9 )
→ ( x² - 12x + y² + 12y + 72 ) = ( x² - 6x + y² - 6y + 18 )
Cancelling the common terms, we get:
→ ( 12y - 12x + 72 ) = ( -6y - 6x + 18 )
→ ( 12x - 6x - 12y - 6y ) = 72 - 18
→ 6x - 18y = 54
Simplifying we get,
→ x - 3y = 9 ...(ii)
Hence solving (i) and (ii) we get,
→ x = 9 + 3y ( From (ii) )
Substituting the value of x in (i) we get:
→ 3 ( 9 + 3y ) + y = 7
→ ( 27 + 9y + y ) = 7
→ 20y = -20
→ y = 20/-20
→ y = -1
Therefore,
→ x = 9 + 3y
→ x = 9 + 3 ( -1 )
→ x = 9 - 3
→ x = 6
Therefore the value of x is 6 and y is -1.
Therefore the center of the circle is ( 6,-1 ).
Radius of the circle:
→ √ [( 6 - 6 )² + ( -1 + 6)²]
→ √ [ 0 + 25 ]
→ √25 = 5 units = r
Hence the radius of the circle is 5 units.
Answer:
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