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____ [38]
3 years ago
6

If (8, −9) and (6, −3) are the endpoints of the diameter of a circle, what is the equation of the circle?

Mathematics
1 answer:
Umnica [9.8K]3 years ago
7 0

(x-7)^{2}  + (y+6)^{2}  = 10 is the equation of the required circle.

Step-by-step explanation:

Step 1 :

Let A = (8,-9) and B = (6,-3) be the endpoints of the diameter of the given circle.

We need to determine the circle's equation with this diameter

Step 2 :

The mid point of the diameter will be center of the required circle.

Let M = (x,y) be the mid point of the circle.

So  we have x = (8+6)÷ 2 , y = ((-9) + (-3)) ÷ 2

M = ( 7, -6) is the midpoint of the diameter and the center of the circle.

Step 3 :

The radius of the circle is  distance between  midpoint and any one of the end point of the diameter that is  distance between  point A(8,-9) and  M ( 7,-6)

Distance between the 2 points (x_{1} ,y_{1})  and  (x_{2} ,y_{2}) is given by

\sqrt{(x_{2}-x_{1})^{2}+ (y_{2}-y_{1})^{2} }

So distance between  A(8,-9) and  M ( 7,-6)  =  \sqrt{(7-8)^{2}+ (-6-(-9))^{2} }

                                                                        =  \sqrt{(1)^{2}+ (3)^{2} } =  \sqrt{10 }

Hence the radius r is \sqrt{10}

Step 4 :

The equation of circle with center (a,b) = (7,-6) and radius r = \sqrt{10} is given by

(x-a)^{2}  + (y-b)^{2} = r^{2}

=> (x-7)^{2}  + (y-(-6))^{2}  = (\sqrt{10} )^{2}

=> (x-7)^{2}  + (y+6)^{2}  = 10 is equation of the required circle.

Step 5 :

Answer :

(x-7)^{2}  + (y+6)^{2}  = 10  is  equation of the required circle.

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a) P(X=2)=(10C2)(0.32)^2 (1-0.32)^{10-2}=0.211

b) P(X> 2)=1-P(X\leq 2)=1-[0.0211+0.0995+0.211]=0.668

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Step-by-step explanation:

1) Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

2) Solution to the problem

Let X the random variable of interest, on this case we now that:

X \sim Binom(n=10, p=0.32)

The probability mass function for the Binomial distribution is given as:

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Where (nCx) means combinatory and it's given by this formula:

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Part a

P(X=2)=(10C2)(0.32)^2 (1-0.32)^{10-2}=0.211

Part b

P(X> 2)=1-P(X\leq 2)=1-[P(X=0)+P(X=1)+P(X=2)]

P(X=0)=(10C0)(0.32)^0 (1-0.32)^{10-0}=0.0211  

P(X=1)=(10C1)(0.32)^1 (1-0.32)^{10-1}=0.0995

P(X=2)=(10C2)(0.32)^2 (1-0.32)^{10-2}=0.211

P(X> 2)=1-P(X\leq 2)=1-[0.0211+0.0995+0.211]=0.668

Part c

P(2 \leq x \leq 5)=P(X=2)+P(X=3)+P(X=4)+P(X=5)

P(X=2)=(10C2)(0.32)^2 (1-0.32)^{10-2}=0.211

P(X=3)=(10C3)(0.32)^3 (1-0.32)^{10-3}=0.264

P(X=4)=(10C4)(0.32)^4 (1-0.32)^{10-4}=0.218

P(X=5)=(10C5)(0.32)^5 (1-0.32)^{10-5}=0.123

P(2 \leq x \leq 5)=0.211+0.264+0.218+0.123=0.816

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