Given 2 circles, find the number of common tangents?

Question

Given 2 circles, C1 centred at (1,0) with radius 1 and C2 with equation  x^2 -12x + y^2+ 20 = 0 . Find the number of common tangents?
A 0
B 1
C 2
D 3
E 4

GMATinsight · Accepted Answer

The answer the question is

Option D

Please find attched image and video for explanation

https://www.youtube.com/watch?v=7NczrYHhdK0

yashikaaggarwal · Answer

Are you sure there are only two tangents? 
because the circles are touching externally
and when two circles touch externally they have 3 common tangents.
one in between and 2 on top and bottom of two circles.
Like this.
Do confirm.

Andrewcoleman · Answer

this should be reclassified out of DS and into PS

EgmatQuantExpert · Answer

Solution

Given

• Two circles C1 and C2.
• C1 is centered at (1,0) and radius of 1
• C2 with an equation  x^2 − 12x +  y^2  + 20=0

To Find

• Number of common tangents of C1 and C2.
 
 Approach and Working Out

•  X^2 − 12X +  Y^2  + 20 =0
 o  X^2  – 2 × 6 × X + 36 +  Y^2  + 20 – 36 = 0
o  X^2  – 2 × 6 × X + 36 +  Y^2 = 16
o  (X – 6)^2  +  Y^2  =  4^2 
o Centered at (6,0) and radius 4.

• Distance between (1,0) & (6,0) is 5 unit and the sum of the radius is also 4 + 1 = 5 unit.
 o It means they will touch externally.
o There will be three common tangents.

• Please note the following points (you should draw to understand this better),
 o If one circle is inside another circle without touching it, then there is no common tangent.
o If one circle is touching another circle from inside, then there is 1 common tangent. (Direct common tangent).
o If two circles intersect each other then there are two common tangents (both direct).
o If two circles touch each other externally then there are 3 common tangents (two direct and 1 transverse)
o If two circles are not touching each other and also do not have any common area then, they have 4 common tangents. (Two direct and two transverse).

Correct Answer: Option C

finisher009 · Answer

Here’s the solution using pattern recognition:
1. Analyze Circle 1: The problem gives you everything you need. Center C1 is (1, 0) and radius r1 is 1.

2. Analyze Circle 2: You need to quickly find its center and radius from the equation x^2 - 12x + y^2 + 20 = 0.
• To find the center, complete the square: (x^2 - 12x + 36) + y^2 = -20 + 36.
• This simplifies to (x - 6)^2 + y^2 = 16.
• So, the center C2 is (6, 0) and the radius r2 is sqrt(16) = 4.

3. Compare the Circles: Now, compare the distance between the centers (d) with the sum of their radii (r1 + r2).
• Distance (d): The distance between (1, 0) and (6, 0) is simply 6 - 1 = 5.
• Sum of Radii (r1 + r2): The sum of the radii is 1 + 4 = 5.

4. Final Conclusion: Since the distance between the centers (d=5) is exactly equal to the sum of the radii (r1+r2=5), the circles touch each other externally at one point. This specific arrangement always has 3 common tangents.

Rules:
• d > r1 + r2: Circles are separate and do not touch. (4 common tangents)
• d = r1 + r2: Circles touch externally. (3 common tangents - this problem’s scenario)
• |r1 - r2| < d < r1 + r2: Circles intersect at two points. (2 common tangents)
• d = |r1 - r2|: Circles touch internally. (1 common tangent)
• d < |r1 - r2|: One circle is completely inside the other. (0 common tangents)

Given 2 circles, find the number of common tangents?

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GMAT Problem Solving (PS) Questions

Given 2 circles, find the number of common tangents?

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