In the rectangular coordinate system, points (4, 0) and

The answer is E.

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.

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

For more check Coordinate Geometry chapter of Math Book: math-coordinate-geometry-87652.html

Hope it helps.
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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) There is no finite maximum value

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

Answer: E.
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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.

Agree with E.

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.

Yep E , we need to know the origin to determine the radius and from the above information we cannot determine the origin.

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.

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.

For more check Coordinate Geometry chapter of Math Book: math-coordinate-geometry-87652.html

Hope it helps.
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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 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.

Responding to a PM

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.

Hope it's clear

Regards
Harsh
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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

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

Hi Bunuel

Could you please show the passages by which you got to the equation of the circle "x2+(y−a)2=42+a2"?

Thanks

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:
\(x^2+y^2=r^2\).

Check more here: https://gmatclub.com/forum/math-coordina ... 87652.html
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I'm totally not able to visualize how making a circle bigger and bigger would still ensure that the circle will keep passing through the two points given. Could you please explain that and also why the centre of the circle will always be on y axis in a bit more detail, im very confused.

Additionally, is there any circle simulations available where I could just punch in numbers to see how the radius of circle differs but it keeps passing through the same points (like the slope simulation that is available on google)

VeritasKarishma Bunuel ScottTargetTestPrep BrentGMATPrepNow
mikemcgarry

Consider the line joining (4, 0) and (-4, 0). Since both its end points lie on the circle, the line is a chord.
Now consider the case when this chord is the diameter of the circle. Then radius of the circle is 4.
But what if the chord is not the diameter but a small chord in the circle? Then its radius will be much greater.



Consider that A is (-4, 0) and B is (4, 0).
The black circle will have a radius of 4. The red circle will have greater radius. The green circle will have even greater radius.
and so on... The chord can be infinitesimally small compared with the circle and hence the radius can be as large as we wish.
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The answer will take 5 secs after reading the question! provided you realize that only 2 points can not constrain a circle, so effectively there are infinite number of circle possible with 2 given points on its circumference!
A little out of context, but to emphasis the point made. If we make 2 points on a paper and put it on the ground, you maybe able to draw a circle but please also note that those 2 points also lie on the circumference of the Earth! So really, no maximum radius is possible!

Where am I going wrong with this:

If the length of chord is 8.
The radius must be joining both end of the Chord and forming a triangle where the two opposite radius will give angles 45, 45, and thus angle at the centre becomes 90. This makes this a 1:1:Root 2 triangle and hence Radius 4 root 2.

Can you please help?
Bunuel
JDLaw
KarishmaB