Question

Point A(-1,y) and B(5,7) lie on a circle with Centre O(2,-3y). Find values of y. Hence find the radius of the circle. ​

Answer 1

Length AO = length BO

As, both are the radius of given circle,

Therefore, By applying distance formula,

(2-(-1))² + (y-(-3y))²= (5-2)² + (7-(-3y))²

=>9+16y² =9 +(7+3y)²

=>16y²=49+9y²+42y

=>7y²-42y-49=0

=>y²-6y-7=0

=>y²-7y+y-7=0

=>y(y-7)+1(y-7)=0

(y-7)=0 or (y+1)=0

y=7 or y=-1

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Answer 2

Answer:

$\huge\boxed{\fcolorbox{red}{pink}{Your's Answer}}$

$\red {QUESTION}$

Point A(-1,y) and B(5,7) lie on a circle with Centre O(2,-3y). Find values of y. Hence find the radius of the circle.

$\green {ANSWER}$

Since O is the centre of the circle and A,B are points on its circumference.

OA=OB=Radius

$\sqrt{( {2 + 1})^{2} + ( { - 3y - y})^{2} } =$

$=> \sqrt{( {2 - 5})^{2} + ( { - 3y - 7})^{2} }$

$=> 9 + {16y}^{2} = 9 + ( {3y + 7})^{2}$

$=> {16y}^{2} = {9y}^{2} + 42y + 49$

$=> {7y}^{2} - 42y - 49 = 0$

$=> {y}^{2} - 6y - 7 = 0$

$=> (y - 7)(y + 1) = 0$

$=> y = - 1</em><em>,</em><em>7$

When y=-1: The coordinates of O,A and B are O(2,3), A(-1,-1) and B(5,7) respectively.

Radius=OA=$\sqrt{( {2 + 1})^{2} + ( {3 + 1})^{2} } = 5$

When y=7: The coordinates O,A and B are O(2,-21), A(-1,7) and B(5,7) respectively.

Radius=OA=$\sqrt{( {2 + 1})^{2} + ( { - 21 - 7})^{2} }$

$= \sqrt{9 + 784}$

$\sqrt{793}$

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