[ASK] A Line Intercepting A Circle

  • MHB
  • Thread starter Monoxdifly
  • Start date
  • #1
Monoxdifly
MHB
284
0
A circle whose center is (2, 1) intercepts a line whose equation is 3x + 4y + 5 = 0 at point A and B. If the length of AB = 8, then the equation of the circle is ...
A. \(\displaystyle x^2+y^2-24x-2y-20=0\)
B. \(\displaystyle x^2+y^2-24x-2y-4=0\)
C. \(\displaystyle x^2+y^2-12x-2y-11=0\)
D. \(\displaystyle x^2+y^2-4x-2y+1=0\)
E. \(\displaystyle x^2+y^2-4x-2y+4=0\)

I don't know how to do it. Judging by the center of the circle, the answer must be either D or E. However, when I checked both of them with Desmos, neither circles even touches the line. How should I do it? Even if there's no right option, I would still like to know in case I encounter this kind of question again.
 

Answers and Replies

  • #2
jonah1
108
0
Beer induced reaction follow.
A circle whose center is (2, 1) intercepts a line whose equation is 3x + 4y + 5 = 0 at point A and B. If the length of AB = 8, then the equation of the circle is ...
A. \(\displaystyle x^2+y^2-24x-2y-20=0\)
B. \(\displaystyle x^2+y^2-24x-2y-4=0\)
C. \(\displaystyle x^2+y^2-12x-2y-11=0\)
D. \(\displaystyle x^2+y^2-4x-2y+1=0\)
E. \(\displaystyle x^2+y^2-4x-2y+4=0\)

I don't know how to do it. Judging by the center of the circle, the answer must be either D or E. However, when I checked both of them with Desmos, neither circles even touches the line. How should I do it? Even if there's no right option, I would still like to know in case I encounter this kind of question again.
https://www.desmos.com/calculator/ngflni4a1s
 
  • #3
Evgeny.Makarov
Gold Member
MHB
2,437
929
Find the distance $h$ from the circle center $O$ to the line using this formula. Then you have an isosceles triangle with base $AB=8$ and height $h$. Find the equal legs of the triangle, which is the radius of the circle.

I don't see the correct answer in any of the variants. I believe the red circle on your sketch is the correct one.
 
  • #4
skeeter
1,104
1
$(x-2)^2 + (y-1)^2 = r^2$

$AB = 8 \implies r >4 \implies r^2 > 16$
 
  • #5
Monoxdifly
MHB
284
0
Okay, thank you for all your answers. :)
 

Suggested for: [ASK] A Line Intercepting A Circle

  • Last Post
Replies
8
Views
923
  • Last Post
Replies
19
Views
267
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
3
Views
276
  • Last Post
Replies
4
Views
621
  • Last Post
Replies
3
Views
593
Replies
4
Views
1K
Replies
10
Views
306
Replies
2
Views
638
  • Last Post
Replies
1
Views
610
Top