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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. - Mathematics

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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.

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Solution

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`

Concept: Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
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