Circle and a line dividing the circumference

  • #1
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Homework Statement


The line "4x + 3y -4=0 " divides the circumference of circle centred at (5,3) in ratio 1:2.whats the equation of the circle

Homework Equations



2πr
Distance of a line from point
Radius of general circle

The Attempt at a Solution


I tried finding the distance of the centre from the chord.
Then i made a relation between radius and common factor of the ratio between the two circumfernce divisions
And now im stuck :(
 

Answers and Replies

  • #3
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you can check out “circle functions” in detail on the internet.
How do circle functions help here ???
They havent given any angle or whats so ever .
 
  • #7
SammyS
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Thank you so much !!!!
What is your result & how did you get it?

(We always like to know after helping someone.)
 
Last edited:
  • #8
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Okay sorry for the delay SammyS.
So heres what i did
 
  • #9
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i used the circle function to relate the two circum ference. ( arc lenght = Radius × Angle in radians) and i got the value for alpha whivh was the angle i asumed.

Then i used angle sum property of a triangle and found out that twas a equilateral triangle of angle 60°(i dont think there are other types of equilaterals in a cartisian plane !!!)

Then i found the perpendicular distance between the centre of circle and the line which came out to be one (using distance formula between line and point)

And then i related radius and that distance by using sin 60° and got radius.

Then by using the radius formula of general circle i found C.

And hence the equation came to be
x2 +y2 -10x -6y +30 = 0

I pray i didnt do a calculation error
 
  • #10
SammyS
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No that's not right, and you didn't give a lot of detail on the individual steps.
i used the circle function to relate the two circumference. ( arc length = Radius × Angle in radians) and i got the value for alpha which was the angle i assumed.
I could ask what you got for the angle, but probably more important: What was the arc length you used to get the angle, and how does that arc length compare to the circumference of the circle?
Then i used angle sum property of a triangle and found out that it was a equilateral triangle of angle 60°(i dont think there are other types of equilaterals in a Cartesian plane !!!)
It's not clear how this triangle is oriented with respect to the circle or to the line.
Then i found the perpendicular distance between the centre of circle and the line which came out to be one (using distance formula between line and point)
What is the equation of the line which is perpendicular to the given line?

That is not the correct distance.
And then i related radius and that distance by using sin 60° and got radius.
What radius did you get?
Then by using the radius formula of general circle i found C.
What is this C?

Give this formula.
And hence the equation came to be
x2 +y2 -10x -6y +30 = 0

I pray i didnt do a calculation error
That is the equation of a circle centered at (5, 3), and its radius is 2.

Is that what you intended?

The graph given by @Young physicist in post (#4) is not necessarily to scale, but ifs clear that when you consider the x and y intercepts of the given line, the radius of the circle needs to be much greater than 2 to have any points in common with the given line.
 
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  • #12
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Just giving a general concept:wink:
This diagram is pretty much on the mark. I would print it out and then draw the line from centre of circle to (calculated) midpoint of the chord. Mark the angles you can obtain from the circumference ratio and the rest should be simple trig!
 
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  • #13
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No that's not right, and you didn't give a lot of detail on the individual steps.
I could ask what you got for the angle, but probably more important: What was the arc length you used to get the angle, and how does that arc length compare to the circumference of the circle?
It's not clear how this triangle is oriented with respect to the circle or to the line.
What is the equation of the line which is perpendicular to the given line?

That is not the correct distance.
What radius did you get?
What is this C?

Give this formula.

That is the equation of a circle centered at (5, 3), and its radius is 2.

Is that what you intended?

The graph given by @Young physicist in post (#4) is not necessarily to scale, but ifs clear that when you consider the x and y intercepts of the given line, the radius of the circle needs to be much greater than 2 to have any points in common with the given line.
Yes i just realised that my distance is wrong.it should be 5.(i forgot to root something)

Wait ill just post a detailed solutiin in a few min
 
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  • #14
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Ps - is it compulsary that i type the full soln or can i just send an attachement with the handwritten working ? Someone had said it was against forum policy to do the later.just confirming
 
  • #15
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Okay so heres what i did.
 
  • #16
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So first i calculated Arc Length
Now the ratio of circumference is 1:2.
Let the common factor be x
S = r θ where θ is central angle of the minor arc
x =r ×θ......
X/θ = r (1)

Lets look at major arc

2X = r ×(2π-θ)
2X= (X/θ)(2π-θ)

2θ=2π-θ
3θ=2π
θ =2π/3

Converting to degrees we get (2 ×180)/3=120°
(Not 60 as i previously screwed up in calculations)
 
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  • #17
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Now lets look at the distance tween cente and the line

This is given by | (ax + by + k)÷(root of (a2 +b2|

Where a b and k are constants in the equation of line and x and y are coardinates of the point
And" | | "represent modulus

With that we get
| (4 ×5 + 3×3 -4) ÷ 5| = 5(not 1 as i previosly did)
 
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  • #18
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If we look at that triangle and use angle sum prop
We get Φ +Φ +120 = 180

Φ=30
 
  • #19
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Hence now we can finaly find radius
By using sin 30 =opposite side÷hypotenuese

(1/2) =(5/r)
r =10
 
  • #20
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With this we can use the formula for radius

g2 + f2 +c = 10
Where -g and -h are coord nates of centre

25 +9 +c =10

c= -24
 
  • #21
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And now that we have all unknows the equation of general circle

x2 +y2 + 2gx +2fy +c =0

Hold on i made an error

The formula for radius is g2 + f2 -c = 10

Hence r = -24
 
  • #22
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Anyway so equation of circle is

x2 +y2 -10x -6y -24 =0
 
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  • #23
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I checked the answer given and its not matching
 
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  • #24
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We r suposed to get
x^2 +y^2 -10x -6y -66 =0
 
  • #25
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We r suposed to get
x^2 +y^2 -10x -6y -66 =0
Any ideas ?
 

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