MGMAT Circles n coordinate

[rommysingh]

In the figure above, the circles are centered at O(0, 0) and P(10, 0). Line AB is tangent to both circles, at points A and B respectively, and intersects the x-axis at point X. What is the x-coordinate of point X ?

(1) The area of the circle centered at point P is 4 times the area of the circle centered at point O.

(2) BX is twice as long as AX.

[Max@Math Revolution]

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Remember equal number of variables and independent equations ensures a solution.




In the figure above, the circles are centered at O(0, 0) and P(10, 0). Line AB is tangent to both circles, at points A and B respectively, and intersects the x-axis at point X. What is the x-coordinate of point X ?

(1) The area of the circle centered at point P is 4 times the area of the circle centered at point O.

(2) BX is twice as long as AX.

In the original condition, we have 2 variables since we need to find out the radius of circle P and circle O. In order to match the number of variables and equations, we need 2 equations and since there is 1 each in 1) and 2), C has high probability of being the answer. Using both 1) and 2) together, the radius of circle P is twice the size of that of circle O. Since 2) is practically the repetition of this information, the conditions are not sufficient. Therefore the answer is E.

For 95% of the questions, if 1)=2) the answer turns out to be D, with the rest of 5% having E as an answer.

Normally for cases where we need 2 more equations, such as original conditions with 2 variable, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer hence using ) and 2) together. (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

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[AdamTooch]

I understand your explanation and I put E for my answer but MGMAT and GMATclub explanations solve the question to D. Since the tangent lines can be drawn as 2 triangles about the circle and X axis, the triangles are similar and as such the ratios of side lengths are equal thus giving a value for x.

[GMATGuruNY]

In the figure above, the circles are centered at O(0, 0) and P(10, 0). Line AB is tangent to both circles, at points A and B respectively, and intersects the x-axis at point X. What is the x-coordinate of point X ?

(1) The area of the circle centered at point P is 4 times the area of the circle centered at point O.

(2) BX is twice as long as AX.

Solutions to circle problems often require the DRAWING OF RADII.
Here, if we draw radii OA and BP, the following figure is yielded:



In the figure above:
Since vertical angles are equal, the two angles labeled "e" are equal.
Since a radius drawn to a tangent line forms a right angle, ∠OAX = ∠XBP = 90.
Since the two angles labeled "e" are equal, and ∠OAX = ∠XBP = 90, the remaining angles in the two triangles -- the two angles labeled "d" -- must also be equal.

Thus, triangle OAX and triangle XBP have the SAME COMBINATION OF ANGLES:
d-e-90.
Triangles with the same combination of angles are SIMILAR.
In similar triangles, sides opposite equal angles are in the SAME RATIO.
Thus:
BP/OA = BX/AX = PX/OX.

Statement 2:
Since BP/OA = BX/AX = PX/OX, and BX = 2(AX), we get:
PX = 2(OX).

Since OP=10, PX + OX = 10.

Substituting PX = 2(OX) into PX + OX = 10, we get;
2(OX) + OX = 10
3(OX) = 10
OX = 10/3.
Thus, the x-cooridinate of X = 10/3.
SUFFICIENT.

Statement 1:
Thus:
π(BP)² = 4 * π(OA)²
BP² = 4(OA)²
BP = 2(OA), implying that PX = 2(OX).
Statement 1 implies the same information as Statement 2.
Since Statement 2 is sufficient, Statement 1 must also be SUFFICIENT.

The correct answer is D.