Circle Geometry with an Intersecting Line

In summary: If the given answers are wrong there is nothing you can do to arrive at them in a correct way. Your answers to the stated problem are correct---end of story.On the other hand, if the stated answer is correct your stated problem must be incorrect; there is no other possibility.
  • #1
Physiona
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OP warned about not using the homework template
I'm requiring help on a circle geometry question I've done.
The line L, has equation of y=0, and intersects the circle with (3,0) and radius of 29. Find the points of intersection.
My working out:
292 = 841
It's centre is 3,0,
Inserting that in circle equation gives (x-3)2+y2 = 841

Solving simultaneously,
(X-3)2 + y2 = 841 (1)
y=0 (2)
Sub (2) into (1)
(X-3)2 + 02 = 841
x2 -6x + 9 = 841
x2-6x+9-841=0
x2-6x-832=0
(X+26)(x-32)=0
X1= -26 and x2= 32

I'm stuck on what to do after this, I think I use process of elimination, however I'm not sure on the equation of y=0, as its confusing me with the zero and no other coefficients. Can someone help?
Thank you.
 
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  • #2
On the line y=0, x can have any value and y is always 0. So at the x values that you calculated, x=-26, and x=32, what y values would put it on the line?

PS. You might want to graph that line and the circle to see if there are easier ways to answer the problem.
 
  • #3
FactChecker said:
On the line y=0, x can have any value and y is always 0. So at the x values that you calculated, x=-26, and x=32, what y values would put it on the line?

PS. You might want to graph that line and the circle to see if there are easier ways to answer the problem.
Wouldn't the y values be 0, to put the x coordinates in the line? I have drawn the graphs, but I still don't get the correct answer.
The mark scheme says it intersects at (24,-20) and (-18,-20) I am not entirely sure where those values are from...
 
  • #4
Physiona said:
Wouldn't the y values be 0, to put the x coordinates in the line? I have drawn the graphs, but I still don't get the correct answer.
The mark scheme says it intersects at (24,-20) and (-18,-20) I am not entirely sure where those values are from...

Either the marking scheme is wrong or you have copied out the question incorrectly. On the line y=0 the intersection with the circle will always have second coordinate = 0.
 
  • #5
Physiona said:
Wouldn't the y values be 0, to put the x coordinates in the line?
Yes. You got the correct answer for the problem that you stated in the original post.
I have drawn the graphs, but I still don't get the correct answer.
If it made your calculated answers obvious, that was all that I was asking for. The graph should confirm your answers.
The mark scheme says it intersects at (24,-20) and (-18,-20) I am not entirely sure where those values are from...
That answer doesn't make sense.
 
  • #6
Physiona said:
The mark scheme says it intersects at (24,-20) and (-18,-20) I am not entirely sure where those values are from...
That would be the correct answer if the line were ##y=-20##.
 
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  • #7
Ray Vickson said:
Either the marking scheme is wrong or you have copied out the question incorrectly. On the line y=0 the intersection with the circle will always have second coordinate = 0.
If I'm honest, it's not an entire correct mark scheme instead it is a solution, so I think the solution itself is wrong, and is confusing me.
 
  • #8
tnich said:
That would be the correct answer if the line were ##y=-20##.

Yes true, however the line is y=0, and the real agony is how I'm meant to actually get the points of intersection.
 
  • #9
FactChecker said:
Yes. You got the correct answer for the problem that you stated in the original post.If it made your calculated answers obvious, that was all that I was asking for. The graph should confirm your answers.That answer doesn't make sense.
Precisely. The solution is incorrect, and I'm not sure where to exactly progress on from there to get the right answer..
 
  • #10
Physiona said:
Precisely. The solution is incorrect, and I'm not sure where to exactly progress on from there to get the right answer..
Sorry, I was not clear. Your answer is correct for the line y=0 and the book answer is wrong . As @tnich stated, the book answer is correct for y=-20.
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  • #11
Physiona said:
Precisely. The solution is incorrect, and I'm not sure where to exactly progress on from there to get the right answer..

What are you trying to say? I cannot understand the point you are making.

If the given answers are wrong there is nothing you can do to arrive at them in a correct way. Your answers to the stated problem are correct---end of story.

On the other hand, if the stated answer is correct your stated problem must be incorrect; there is no other possibility.
 
  • #12
FactChecker said:
Sorry, I was not clear. Your answer is correct for the line y=0 and the book answer is wrong . As @tnich stated, the book answer is correct for y=-20.
View attachment 221817
So my answer is correct. I'm just missing the y values though would they count as zero?
 
  • #13
Ray Vickson said:
What are you trying to say? I cannot understand the point you are making.

If the given answers are wrong there is nothing you can do to arrive at them in a correct way. Your answers to the stated problem are correct---end of story.

On the other hand, if the stated answer is correct your stated problem must be incorrect; there is no other possibility.
No you don't understand my point. I've worked out the x values, I'm missing the y values, due to the line equation of y=0.
 
  • #14
Physiona said:
So my answer is correct. I'm just missing the y values though would they count as zero?
Yes, the points in your answer are on the y=0 line. So (-26,0) and (32,0) are correct for the problem that you stated.
 
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  • #15
If y=0 the problem seems rather too easy. It really looks like a typo and it should be y=-20.
 
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Related to Circle Geometry with an Intersecting Line

1. What is a circle in geometry?

A circle is a shape that is formed by a set of points that are all equidistant from a given center point. It can also be defined as the locus of all points equidistant from a given point.

2. What is the equation for a circle?

The equation for a circle is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center point and r is the radius.

3. How does an intersecting line affect a circle in geometry?

When a line intersects a circle, it creates two points of intersection. These points can be used to find the tangent of the circle at that point, as well as the length of the chord created by the intersecting line.

4. What is the relationship between the intersecting line and the circle's radius and diameter?

The intersecting line is perpendicular to the radius of the circle at the point of intersection. It also bisects the diameter of the circle, creating two equal segments.

5. How can we use circle geometry with intersecting lines in real-world applications?

Circle geometry with intersecting lines has many real-world applications, such as in engineering to design round structures or in navigation to determine the shortest route between two points. It is also used in physics and astronomy to calculate the motion of objects in circular orbits.

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