Does the curve \((x-a)^2+(y-b)^2=16\) intersect the Y-axis? GMAT Data Sufficiency

Question: Does the curve \((x-a)^2+(y-b)^2=16\)  intersect the Y-axis?

1. \(a^2+b^2>16\)
2. a = |b| + 5

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“Does the curve \((x-a)^2+(y-b)^2=16\) intersect the Y-axis?”– is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken f0rom the book "GMAT Quantitative Review".
The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation

Approach Solution 1:

There is only one approach solution to this problem.

In an x-y Cartesian coordinate system, the circle with centre (a,b) and radius r is the set of all points (x,y) such that:

\((x-a)^2+(y-b)^2=r^2\)

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram above, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x-a and x-b.

Curve of \((x-a)^2+(y-b)^2=16\) is a circle centered at the point (a,b) and has a radius \(\sqrt{16}=4\) .

Now, if a, the x-coordinate of the centre, is more than 4 or less than -4 then the radius of the circle, which is 4, won’t be enough for the curve to intersect with the Y-axis.

O basically the question asks whether |a| > 4: if it is, then the answer will be NO: the curve does not intersect with Y-axis and if it’s not, then the answer will be YES: the curve intersects with Y-axis.

1. \(a^2+b^2>16\) . Clearly this statement is insufficient as |a| may not be more than 4
2. a = |b| + 5. As the least value of the absolute value (in our case |b|) is zero then the least value of a will be 5, so in any case |a| > 4, which means that the circle does not intersect the Y-axis.

Hence, this statement is sufficient.

Correct Answer: B

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