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# Two circles, with centers at points (3, 7) and (−1, 4) and with radii

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Math Expert
Joined: 02 Sep 2009
Posts: 65807
Two circles, with centers at points (3, 7) and (−1, 4) and with radii  [#permalink]

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01 Jun 2020, 08:27
00:00

Difficulty:

65% (hard)

Question Stats:

40% (02:10) correct 60% (02:04) wrong based on 43 sessions

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Two circles, with centers at points (3, 7) and (−1, 4) and with radii 2 units and 7 units respectively, are drawn in the x-y plane. What is the number of common tangents that the circles can have ?

A. 0
B. 1
C. 2
D. 3
E. 4

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Joined: 03 Jun 2019
Posts: 3362
Location: India
GMAT 1: 690 Q50 V34
WE: Engineering (Transportation)
Re: Two circles, with centers at points (3, 7) and (−1, 4) and with radii  [#permalink]

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01 Jun 2020, 20:17
1
1
Bunuel wrote:
Two circles, with centers at points (3, 7) and (−1, 4) and with radii 2 units and 7 units respectively, are drawn in the x-y plane. What is the number of common tangents that the circles can have ?

A. 0
B. 1
C. 2
D. 3
E. 4

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Given: Two circles, with centers at points (3, 7) and (−1, 4) and with radii 2 units and 7 units respectively, are drawn in the x-y plane.
Asked: What is the number of common tangents that the circles can have ?

Distance between the centers =$$\sqrt{(3+1)^2 + (7-4)^2} = 5$$

Attachment:

Screenshot 2020-06-02 at 9.46.45 AM.png [ 20.42 KiB | Viewed 437 times ]

Since the distance between centers = 5 = 7 - 2

Number of common tangents = 1

IMO B
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Kinshook Chaturvedi
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Re: Two circles, with centers at points (3, 7) and (−1, 4) and with radii  [#permalink]

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01 Jun 2020, 11:16
IMO 1
The distance between 2 points = => √(y2-y1)^2+(x2-x1)^2
=> √(7-4)^2+{3-(-1)}^2
=> √(3)^2+(4)^2
=> √9+16 => √25 = 5 units.
The radius of one circle A is 7 and other B is two.
=> the circle B with the centre (3,7) will extend to 2 more units to Z. Making the whole distance from point (-1,4) to the Z will be 7
Circle A and B coincided with each other at point Z
=> There is only one tangent possible from a given point

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Intern
Joined: 28 Mar 2018
Posts: 44
Re: Two circles, with centers at points (3, 7) and (−1, 4) and with radii  [#permalink]

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01 Jun 2020, 11:41
Bunuel wrote:
Two circles, with centers at points (3, 7) and (−1, 4) and with radii 2 units and 7 units respectively, are drawn in the x-y plane. What is the number of common tangents that the circles can have ?

A. 0
B. 1
C. 2
D. 3
E. 4

Distance between the centers (d) = 5
As d < 2 + 7, the circles should intersect at two distinct points.

Hence, they can have 2 common tangents (option C)

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IESE School Moderator
Joined: 11 Feb 2019
Posts: 311
Re: Two circles, with centers at points (3, 7) and (−1, 4) and with radii  [#permalink]

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03 Jun 2020, 12:39
IMO B

For the circle with Centre (3,7) and radius 2, one extreme x-axis value =5
Another circle with Centre (-1,4) and radius 7, one extreme x-axis value =5

==> 1st circle is inner circle of 2nd one with both touching at x=5.

So one tangent will be common to both.
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Cheers,
NJ
Re: Two circles, with centers at points (3, 7) and (−1, 4) and with radii   [#permalink] 03 Jun 2020, 12:39