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In the rectangular coordinate system, points (4, 0) and (– 4, 0) both lie on circle C. What is the maximum possible value of the radius of C ?

(A) 2 (B) 4 (C) 8 (D) 16 (E) None of the above

I'm getting E

It can be B, but the points mentioned can be a chord and that would make the radius larger. I'm getting other calculations but none are available or can't be determined.

It takes 3 distinct points to define a circle. Only 2 are given here.

The two points essentially identify a single chord of the circle C. Since no other information is provided, however, the radius of the circle can essentially be anything. All this information tell us is that the radius is greater than 4. It does not give us an upper limit.

Another way to look at it is that the two points, lets call them A and B, are equidistant to the centre of the circle, lets call that O. i.e. OA = OB Hence the centre will lie on the Y axis (anywhere where x = 0). So not enough information to determine.

Yet another way to look at it is: Radius^2 = (Difference of X of O to A)^2 + (Difference of Y of O to A)^2 From the question stem we know that A is (4,0). Using the above logic we also know that the centre lies on x=0. Using B would yield the same result as we are after distance it will always end up being positive anyway. This formula reduces to (4-0)^2 + (y-0)^2 = R^2 Depending on the value of y, the length of the radius will keep growing.

The only thing we can conclude from the question that center lies on the Y-axis. But it could be ANY point on it, hence we can not determine maximum value of r.
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The question is more like a DS question. Rephrase it and you will get "If two points given are enough to define the maximum possible radius of the circle?" The answer is no, cause the radius could be as low as 4 if the points are at the maximum distance from the center and the line between them is the diameter or the radius could be infinitely large if the line between the points is the chord.

The only thing we can conclude from the question that center lies on the Y-axis. But it could be ANY point on it, hence we can not determine maximum value of r.

Can we also conclude that the points (4,0) and (-4,0) lie in first and 2nd quadrant so with that we cannot calculate the distance between two points ( which will be radius of circle ) ; because in order to calculate distance we need points in opposite direction. So if the points were in Ist and 3rd quadrant we could have calculated the distance

The only thing we can conclude from the question that center lies on the Y-axis. But it could be ANY point on it, hence we can not determine maximum value of r.

Can we also conclude that the points (4,0) and (-4,0) lie in first and 2nd quadrant so with that we cannot calculate the distance between two points ( which will be radius of circle ) ; because in order to calculate distance we need points in opposite direction. So if the points were in Ist and 3rd quadrant we could have calculated the distance

I think you are a little bit confused here.

You CAN calculate the distance between any two points with given coordinates on a plane (no matter in which quadrants they are). For example the distance between two points (4,0) and (-4,0) is simply 8.

Generally the formula to calculate the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\).

Next, the distance between (4,0) and (-4,0) won't necessarily be the DIAMETER of a circle. The minimum length of a diameter is indeed 8 (so min r=4) but as ANY point on the y-axis will be equidistant from the given points then any point on it can be the center of the circle thus the maximum length of the radius is not limited at all.

if we join the line connecting the the points (-4,0) and (4,0) to the center of the circle say (0,y), radius will be maximum at the point where the area formed by the above triangle is min. The area will be 0 if the height is 0, which means the center is in the line connecting two pnts (-4,0) and (4,0). Isn't?

if we join the line connecting the the points (-4,0) and (4,0) to the center of the circle say (0,y), radius will be maximum at the point where the area formed by the above triangle is min. The area will be 0 if the height is 0, which means the center is in the line connecting two pnts (-4,0) and (4,0). Isn't?

The red part is not correct.

Again: The minimum length of a diameter is indeed 8 (so min r=4) but as ANY point on the y-axis will be equidistant from the given points then any point on it can be the center of the circle thus the maximum length of the radius is not limited at all.

Check 2 possible circles: Circle with min radius of 4 (equation x^2+y^2=4^2):

Attachment:

radius 4.gif [ 3.22 KiB | Viewed 23411 times ]

Circle with radius of 5 (equation x^2+(y-3)^2=5^2):

Attachment:

radius 5.gif [ 2.78 KiB | Viewed 23408 times ]

Generally circle passing through the points (4, 0) and (– 4, 0) will have an equation \(x^2+(y-a)^2=4^2+a^2\) and will have a radius of \(r=\sqrt{4^2+a^2}\). As you can see min radius will be for \(a=0\), so \(r_{min}=4\) and max radius is not limited at all (as \(a\) can go to +infinity as well to -infinity).

[quote="Mongolia2HBS"]Only 2 points are given, so E.[/quo

suppose the question says that what is the maximum possible radius of the among the given choices then we can try out the highest choices and foind out whether the the given choice satisfies the pythagoras eqn. so that way D is best.

The only thing we know is that Points A(4,0) and B(-4,0) are on circumference. But that does not necessarily mean that they are opposite ends of diameter.

If they are opposite ends of diameter, Radius will be 4, but if they are opposite ends of circle's smallest chord then Radius would be far more greater then the values mentioned in Options. Hence Choice E is correct.
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Query:Hi Harsh, In the above problem I can’t think of a situation where the center will not be the origin, but as per the solution this is not true, Will be great if you can elucidate with a diagram.

Solution: The question tells us about two points on the circumference of a circle and ask us about the maximum possible radius of the circle. There can be a situation where the points are the diameter of the circle or any chord of the circle. Refer the two diagrams below:

In the first diagram we have a situation where AB is the diameter of the circle with the centre at origin. In this case the radius of the circle would be 4.

In the second diagram we can see that AB is a chord of the circle. In such a case we can't say anything about the centre of the circle as well as the radius of the circle.

The question is asking us about the maximum length of the radius but we do not have enough information to calculate the length of the radius. Had the question asked us about the minimum length of the radius, we could have said min(radius) = 4 when AB would be the largest chord of the circle i.e. the diameter.

Re: In the rectangular coordinate system, points (4, 0) and [#permalink]

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10 Apr 2016, 12:08

if we solve the question blindly then the answer appears to be option B ( which will definitely be the minimum radius ) but as the question asks for maximum radius given line between (4,0) and (-4,0) can be just a chord with a even bigger radius.the only thing we can conclude is that the center of the circle lies on y -axis for sure. therefore correct answer option E

Re: In the rectangular coordinate system, points (4, 0) and [#permalink]

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10 Apr 2016, 12:25

In the rectangular coordinate system, points (4, 0) and (– 4, 0) both lie on circle C. What is the maximum possible value of the radius of C ?

(A) 2 (B) 4 (C) 8 (D) 16 (E) None of the above

Equation of circle is (X-a)^2+ (Y-b)^2 = R^2 where a and b are the center points and R is the radius of the circle

so in order to find a, b and R we need a atleast of 3 points, which is not that case, however as both (4,0) and (-4,0) lies on the circle we can say that circle is symmetric about y axis, but nothing more can be said about it
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