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Answers
Step-by-step explanation:
Solutions :-
a)
Centre of the circle = Fountain
Coordinates of the fountain =O (2a-1,7)
Point on a circle = A(-3,-1)
Distance between them = Radius of the circle = OA
We know that
Points (x1, y1) and (x2, y2) then the distance between them is √[x2-x1)^2+(y2-y1)^2]
we have
(x1, y1)=(2a-1,7)=>x1=2a-1 and y1 = 7
(x2, y2)=(-3,-1)=>x2=-3 and y2=-1
OA =√[(-3-(2a-1))^2+(-1-7)^2]
=>OA=√[(-3-2a+1)^2+(-8)^2]
=>OA = √[(-2-2a)^2+(-8)^2]
=>OA = √(4+8a+4a^2+64)
=>OA = √(4a^2+8a+68)
But, According to the given problem
Radius = 10 units
=>OA = 10 units
=>√(4a^2+8a+68) = 10
On squaring both sides
=>[√(4a^2+8a+68) ] ^2= (10)^2
=>(4a^2+8a+68) = 100
=>4a^2+8a+68-100=0
=>4a^2+8a-32 = 0
=>4(a^2+2a-8)=0
=>a^2+2a-8=0/4
=>a^2+2a-8=0
=>a^2+4a-2a-8=0
=>a(a+4)-2(a+4)=0
=>(a+4)(a-2)=0
=>a+4 = 0 or a-2 = 0
a = -4 and a = 2
Answer:-
The possible values of a are (-4,2)
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b)
we have A(-3,-1)
and OA = OB = Radius
OA = 10 units
OB = 10 units
if we join AB we get a right angled triangle
By Pythagoras theorem
AB^2=OA^2+OB^2
=>AB^2 = 10^2+10^2
=>AB^2 = 100+100
=>AB^2 = 200
=>AB=√200
=>AB=√(2×100)
=>AB=10√2 units
Answer:-
The value of AB = 10√2 units
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c)
Consider a point on x - axis is A (x,0)
if a= 2 then O(2(2)-1,7)=O(3,7)
Let (x1, y1)=(x,0)=>x1=x and y1=0
Let (x2, y2)=(3,7)=x2=3 and y2=7
OA = 10 units (given)
Points (x1, y1) and (x2, y2) then the distance between them is √[x2-x1)^2+(y2-y1)^2]
=>√[(x-3)^2+(0-7)^2] = 10
=>√[(x-3)^2+(-7)^2]=10
=>√[(x-3)^2+49] = 10
On squaring both sides then
=>(x-3)^2+49 = 100
=>(x-3)^2 = 100-49
=>(x-3)^2 = 51
=>x-3 = ±√51
=>x = 3±√51
Answer:-
The possible value of x = 3±√51
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d)
Consider a point B which lies on y-axis then the coordinates of B =(0,y)
and a= 2 then O(3,7)
Let (x1, y1)=(0,y)=>x1=0 and y1=y
Let (x2, y2)=(3,7)=>x2=3 and y2 = 7
OB = 10 units (given)
Points (x1, y1) and (x2, y2) then the distance between them is √[x2-x1)^2+(y2-y1)^2]
=>√[(3-0)^2+(7-y)^2] =10
On squaring both sides
=>[(3)^2+(7-y)^2]=100
=>9+(7-y)^2 =100
=>(7-Y)^2 =100-9
=>(7-y)^2 =91
=>7-y = ±√91
=>y= -7±√91
Answer:-
The possible value of y = -7±√91
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e)
We join A and B then we get a right angled triangle
∆OAB is a right angled triangle
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Used formulae:-
- Points (x1, y1) and (x2, y2) then the distance between them is
√[x2-x1)^2+(y2-y1)^2] units
- The distance between the centre and any point on the circumference of the circle is called its radius.
- All radii in a circle are equal.
- The square of the hypotenuse is equal to the sum of squares of the other two sides is called Pythagoras theorem.