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THEORY In an x-y Cartesian coordinate system, the circle with center (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 y-b.

If the circle is centered at the origin (0, 0), then the equation simplifies to: \(x^2+y^2=r^2\)

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

Curve of \((x - a)^2 + (y - b)^2 = 16\) is a circle centered at the point \((a, \ b)\) and has a radius of \(\sqrt{16}=4\). Now, if \(a\), the x-coordinate of the center, is more than 4 or less than -4 then the radius of the circle, which is 4, won't be enough for curve to intersect with Y axis. So 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 insufficient as \(|a|\) may or may not be more than 4.

(2) \(a = |b| + 5\) --> as the least value of 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. Sufficient.

You guys have to tell me what exactly is not clear in my explanation above.

You have to keep in mind that we are dealing with an equation of a circle with radius 4. \(a\) places the center of this circle closer to or farther from the Y axis. From S2 we know that \(a \ge 5\), so the circle does not intersect the Y axis.

THEORY In an x-y Cartesian coordinate system, the circle with center (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 y-b.

If the circle is centered at the origin (0, 0), then the equation simplifies to: \(x^2+y^2=r^2\)

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

Curve of \((x - a)^2 + (y - b)^2 = 16\) is a circle centered at the point \((a, \ b)\) and has a radius of \(\sqrt{16}=4\). Now, if \(a\), the x-coordinate of the center, is more than 4 or less than -4 then the radius of the circle, which is 4, won't be enough for curve to intersect with Y axis. So 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 insufficient as \(|a|\) may or may not be more than 4.

(2) \(a = |b| + 5\) --> as the least value of 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. Sufficient.

Answer: B.

YOUR explanation is very detailed and appreciable.

consider stmnt 2. take a case as a=2 & b=3 or b=3 & a=2. both cond is in agreement with the statement 2. so the eq will be (x-2/3)^2+(x-3/3)^2=16 this circle doesn't intersects the Y axis anywhere.

so the final ans will be E.
_________________

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Attitude determine everything. all the best and God bless you.

YOUR explanation is very detailed and appreciable.

consider stmnt 2. take a case as a=2 & b=3 or b=3 & a=2. both cond is in agreement with the statement 2. so the eq will be (x-2/3)^2+(x-3/3)^2=16 this circle doesn't intersects the Y axis anywhere.

so the final ans will be E.

First of all, OA (the final answer) for this question is B, not E.

Next, your examples (the red part) is not valid as if \(a=2\) and \(b=3\) then \(2\neq{|3|+5=8}\) or if \(a=3\) and \(b=2\) then \(3\neq{|2|+5=7}\) so these values of \(a\) and \(b\) do not satisfy statement (2).

Moreover, if \(a=2\) and \(b=3\) then circle with equation \((x - a)^2 + (y - b)^2 = 16\) does intersect Y-axis (so your conclusion is also wrong, because the formula you wrote is wrong).

Again the circle given by \((x-a)^2+(y-b)^2=r^2\) has center at point (a, b) and radius r, so if \(a=2\) and \(b=3\) then circle given by \((x - 2)^2 + (y - 3)^2 = 16\) has its center at (2,3) and has a radius of 4 and it intersects Y axis at points \((0, \ 3-2\sqrt{3})\) and \((0, \ 3+2\sqrt{3})\). You can see it on the below diagram:

Attachment:

MSP30219de6acg5dc779ic0000351eaf739i21c8f1.gif [ 3.04 KiB | Viewed 9829 times ]

Bunuel, I donno if u r still on these forums, but I need help with this :

" so in any case , which means that the circle does not intersect the Y axis."

I understood everything else except this statement.

\(|a|>4\) i.e either a>4 OR a<-4. If a, the x coordinate of the center of the circle, is more than 4 units away from the y-axis, the circumference of the circle will NEVER intersect the y-axis because the radius of the circle is 4 units. Thus, if we know that a is indeed >4 or <-4, we will definitely know that the circle doesn't intersect y-axis.
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(x-a)^2+(y-b)^2 = 16 is a circlce with centre(a,b) and radius 4.

(1)=>distance of (a,b) from origin in the coordinate system is>4. There are multiple circles with their center on the x-axis which will not intersect y-axis. Then again there are circles with center on the y-axis which will intersect y-axis. hence, not sufficient. (2)=>a=modb+5 Thus, the circle has center at (5+b,b) or (5-b,b) I solved the equn of circle with the y-axis (x=0). we get y-b = sqrt(b^2-10b-9) or sqrt(b^2+10b-9) in any case it is possible to have imaginary solution for some values of b and not others. hence, not sufficient.

you are right! The center of the circle is atleast 5 units away from the origin; so no intersection with x=0 is possible. I wonder why I could not derive it from my equation solving though.

GMAT TIGER wrote:

bigfernhead wrote:

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

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

I have no idea how to tackle this... please explain. Thanks.

Agree with E.

Equation of the circle: (x-a)^2 + (y-b)^2 = 16 Origin = (a, b) & redius of this circle = 4

The values of (a, b) doesnot decide the position of circle. they are x and y along with a and b determin the circle's location.

We need to know 1: a^2 + b^2 > 16 a and b could be anything because their squares are +ve and > 16.

2: a = lbl + 5. now we know that: a is +ve and is > 5 but b? we do not know. it (b) could be anything.

1&2: are we getting anything extra bit of infromation from 2 on 1 or vice versa. No.

I totally agree with the explanation below. The Y axis is at X=0. Since from S2 \(a \ge 5\), we can be sure that X coordinate of the center of the circle will be 5 points away from the Y axis. If the radius of the circle is 4 points, we are sure that the circle does NOT intersect the Y axis. We should not be concerned with what the value of \(b\) is as we have to check only the intersection with the Y axis. We would have to check the value of \(b\) only if the question asked for the intersection with the X axis where Y=0.

The "curve" here refers to the equation of the circle.

I hope the explanation makes sense. The OA is B.

nfernandes wrote:

I am in doubt. from 2) a= |b|+5

I take that to mean a is always positive and >= 5. With a radius of 4 and x coordinate >=5 it cannot intersect the y axis.