Finding the Length of a Chord on a Circle

In summary, the chord cuts off the circle from the straight line x-y+2+=0. The equation of the new circle is x^2+y^2+2gx+2fy+c=0. The length of the chord is 4(2)^1/2.
  • #1
Dumbledore211
111
1

Homework Statement


Find the length of the chord which the circle 3x^2+3y^2-29x-19y+56=0 cuts off from the straight line x-y+2+=0. Find the equation of the circle with this chord as diameter


Homework Equations


x^2+y^2+2gx+2fy+c=0


The Attempt at a Solution


I can solve the second part of the question very easily. What I am really finding difficult is trying to construct a method of calculating the length of the chord. Is there any formula or equation for it?
 
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  • #2
Dumbledore211 said:

Homework Statement


Find the length of the chord which the circle 3x^2+3y^2-29x-19y+56=0 cuts off from the straight line x-y+2+=0. Find the equation of the circle with this chord as diameter


Homework Equations


x^2+y^2+2gx+2fy+c=0


The Attempt at a Solution


I can solve the second part of the question very easily. What I am really finding difficult is trying to construct a method of calculating the length of the chord. Is there any formula or equation for it?

Think about it. If you were dealing with simple geometry where you were given the radius of circle and the distance of chord from the centre, how do you find the length of chord?
 
  • #3
Pranav-Arora said:
Think about it. If you were dealing with simple geometry where you were given the radius of circle and the distance of chord from the centre, how do you find the length of chord?
With the information given, it might be easier just to find the intercepts.
 
  • #4
@haruspex I don't precisely get what you are trying to put across. Are you suggesting that I should find the intercepts of the straight line as well as the radius of the circle from the given two equations. Tell me how the intercepts of the straight line relate with the radius of the circle??
 
  • #5
@Pranav Arora But the distance of chord from the centre is not given..
 
  • #6
Dumbledore211 said:
@Pranav Arora But the distance of chord from the centre is not given..

You can find it...You have the equation of chord and coordinate of center.
 
  • #7
Dumbledore211...Simply find the points of intersection of the circle with the straight line .This will give you two points in the plane .In the first part you have to find the length of the chord which is nothing but the distance between these two points .
 
  • #8
Dumbledore211 said:
@haruspex I don't precisely get what you are trying to put across. Are you suggesting that I should find the intercepts of the straight line as well as the radius of the circle from the given two equations. Tell me how the intercepts of the straight line relate with the radius of the circle??

If you have the intercepts you can use it to find the position of center of the new circle. ED- And the length as Tanya pointed out above(crossed posts)
 
  • #9
Thank you, Tanya Sharma. I finally got the answer which is 4(2)^1/2
 
  • #10
Dumbledore211 said:
Thank you, Tanya Sharma. I finally got the answer which is 4(2)^1/2

Well done...
 

FAQ: Finding the Length of a Chord on a Circle

What is a chord on a circle?

A chord on a circle is a line segment that connects two points on the circle's circumference. It is the longest distance between two points on the circle.

How do you find the length of a chord on a circle?

To find the length of a chord on a circle, you can use the Pythagorean theorem or the chord length formula. The Pythagorean theorem states that the square of the length of a chord is equal to the difference between the square of the radius and the square of the distance from the center of the circle to the chord. The chord length formula is c = 2 * r * sin(a/2), where c is the length of the chord, r is the radius of the circle, and a is the central angle subtended by the chord.

What is the relationship between the length of a chord and the radius of a circle?

The length of a chord and the radius of a circle are inversely proportional. This means that as the length of the chord increases, the radius decreases, and vice versa. This relationship is described by the formula c = 2 * r * sin(a/2), where c is the length of the chord, r is the radius of the circle, and a is the central angle subtended by the chord.

Can you find the length of a chord if you know the diameter of a circle?

Yes, the length of a chord can be found if you know the diameter of a circle. The diameter is equal to twice the radius, so you can use the formula c = 2 * r * sin(a/2) and substitute 2r for d (diameter) to find the length of the chord.

What is the significance of finding the length of a chord on a circle?

Finding the length of a chord on a circle is useful in many real-world applications, such as construction, engineering, and navigation. It allows us to determine the distance between two points on a circle and can help us calculate the size and shape of objects that are circular or have circular components. It is also an important concept in geometry and trigonometry.

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