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प्रश्न
Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y). Find the values of y. Hence find the radius of the circle.
उत्तर
A and B are the two points that lie on the circle and O is the centre of the circle.
Therefore, OA and OB are the radii of the circle.
Using the distance formula, we have:
`OA=sqrt((-1-2)^2+(y+3y)^2)=sqrt(9+16y^2)`
`OB=sqrt((5-2)^2+(7+3y)^2)=sqrt(9+(7+3y)^2)`
Now, OB = OA (Radii of the same circle)
`sqrt(9+(7+3y)^2)=sqrt(9+16y^2)`
9+(7+3y)2=9+16y^2 (squaring both the sides)
49+9y2+42y=16y2
⇒7y2−42y−49=0
⇒y2−6y−7 =0
⇒y2−7y+y−7=0
⇒(y−7)(y+1)=0
⇒y−7=0 or y+1=0
⇒y=7 or y=−1
When y = 7:
Radius of the circle, `OA=sqrt(9+16y^2)=sqrt(9+16×49)=sqrt(793)`
When y = −1:
Radius of the circle, `OA=sqrt(9+16y^2) =sqrt(9+16×1)=sqrt(25)=5`
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