If the chord joining the points (2, -1),(1,
-2) subtends a right angle at he centre of
the circle, then its centre and radius are
Answers
Step-by-step explanation:
The arc of a circle subtending a right angle at any point of the circle in its alternate segment is a semicircle. ○ If a line segment joining ...
Given : chord joining the points (2, -1),(1,-2) subtends a right angle at the centre of the circle
To find : centre and radius of circle
Solution:
Let say Radius = r
& center = h , k
=> (h - 2)² + ( k -(-1))² = r² => ( h - 2)² + ( k + 1)² = r²
& (h - 1)² + ( k -(-2))² = r² => ( h - 1)² + ( k + 2)² = r²
chord joining the points (2, -1),(1,-2) subtends a right angle at he centre of
the circle,
Applying Pythagorean theorem
r² + r² = ( 2 - 1)² + (-1 - (-2))²
=> 2r² = 1 + 1
=> r² = 1
Hence radius = 1
( h - 2)² + ( k + 1)² = ( h - 1)² + ( k + 2)²
=>h² -4h + 4 + k² + 2k + 1 = h² -2h + 1 + k² + 4k + 4
=> -4h + 2k = -2h + 4k
=> -2k = 2h
=> -k = h
( h - 2)² + ( k + 1)² = r²
=> (-k - 2)² + ( k + 1)² = 1²
=> k² + 4 + 4k + k² + 1 + 2k = 1
=> 2k² + 6k + 4 = 0
=> k² + 3k + 2 = 0
=> k² + k + 2k + 2 = 0
=> (k + 2)(k + 1) = 0
=> k = - 2 or - 1
=> h = 2 , 1
Center = ( 2, - 2) or ( 1 , - 1)
Radius = 1
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