What are the coordinates of point A?

Just to illustrate:



Hope it helps.


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The answer is [C]. The approach would be using the statements given.

We are to find the co-ordinates of point A. Hence they have to be a unique set of {x,y}.
Now Statement 1 states, A is 2 units away from (3,4). So basically all points at a distance of 2 units from (3,4) can be A. Hence the locus of Point A is a circle with the equation: \((x-3)^2 + (y-4)^2 = 9\). Clearly Statement 1 alone is not sufficient.

Now Statement 2 states A is 3 units away from (0,0). Similarly it can said that A can be a circle with its center on (0,0) with a radius of 2. Equation: \(x^2 + y^2 = 4\). Again its insufficient alone.

As we combine them we know that two circles can intersect at two points of maybe just touch at one! A would be plausible only if the circles meet at one point. Now if the circles met at one point the distance between the centers = sum of the radii. Distance between centers = 5 units and clearly sum of radii is 5! Hence we can tell for sure that the circles touch at one point which is A. Therefore the answer is [C].

Hope it helps!

Regards,
A

Thanks! Just to confirm, the equation of the circles from statements 1 and 2 respectively would be :-

1) \((x-3)^2 + (y-4)^2 = 4\)
2) \(x^2 + y^2 = 9\)

Yes.

In an xy 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: \(x^2+y^2=r^2\)

For more check: math-coordinate-geometry-87652.html

Hope it helps.
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Statement 1: A is 2 units away from (3, 4).
A can be anywhere on the circumference of a circle centered (3, 5) with Radius of 2 units, Hence
NOT SUFFICIENT


Statement 2: A is 3 units away from (0, 0)
A can be anywhere on the circumference of a circle centered (0, 0) with Radius of 3d units, Hence
NOT SUFFICIENT

Combining the two statements
There will be One point of Intersection of Circles drawn by the two statements (As shown in figure), hence
SUFFICIENT

Answer: Option C

A=(x,y) = ?

Per statement 1, coordinates at a distance of 2 units from (3,4) lie on the circumference of a circle with center (3,4) and radius 2 units. Thus all the points on the circumference of this circle can be A's coordinates. Thus this statement is not sufficient.

Per statement 2, coordinates at a distance of 3 units from (0,0) lie on the circumference of a circle with center (0,0) and radius 3 units. Thus all the points on the circumference of this circle can be A's coordinates. Thus this statement is not sufficient.

Combining, we see that the radii of the circles = 2+3 = 5 = distance between the centers of the 2 circles (= ((3-0)^2+)4-0)^2)^0.5) ----> The circles intersect each other at only 1 point. Thus we get a unique coordinate value for A---> C is the correct answer.
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800score Official Solution:

All the points that lie at some specific distance from a given one are situated on the circle, which center is the given point and the radius is the distance. Therefore each statement by itself defines a circle (infinitely many points). So each statement by itself is NOT sufficient.

If we use the both statements together, we have two circles. Point A belongs to each one of them. Two circles can cross each other in one or two points. (Two circles can also have no common points at all, but it is NOT our case). The distance between the centers, (0, 0) and (3, 4), is √(3² + 4²) = √25 = 5. It equals to the sum of the radii, 2 + 3 = 5. Therefore the circles touch each other (cross) in exactly ONE point. Clearly, the coordinates of this point can be found by solving the system of the two equations of the circles. However we do NOT need to actually calculate the coordinates.

Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient.

The correct answer is C.
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Thank you so much Bunuel

when we say sum of radii , does that mean , sum of distance from center to point A(given 2 and 3 in stmt i and ii respectively).
Also lets say if the sum was less than this distance , does that mean circle A can be 2 distinct points?
In that scenario how do we find possible values of A.
Then will these stmts be sufficient
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Hi

Let me attempt to answer your questions.



Yes. The set of all points at a distance of x from another given point O(x,y) forms a circle with centre O and radius x. Therefore, in the given question, the set of all possible values of A at a distance of 2 units from (3,4) forms a circle with centre (3,4) and radius of 2. Similarly, the set of all possible values of A at a distance of 3 units from (0,0) forms a circle with centre (0,0) and radius of 3. The sum of the radii, therefore, is the sum of the distances from the two centres to point A ie; 2+3=5.



It is unclear what you mean by "this distance". The sum of the radii is the sum of the distances of A from the given points. Therefore, the sum of these two distances will always be equal to the sum of the radii.

Alternatively, if you mean the distance between the centres [(3,4) and (0,0)], then the reasoning would be as follows.

If the sum of the radii was less than the distance between the centres, it would mean that the circles would never intersect. Hence, point A can have no possible co-ordinates satisfying both (i) and (ii). We also know neither (i) nor (ii) alone are sufficient. Hope this helps.
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There seems to be a big mistake in the graphical representation.
Radius of red circle is 3 to distance from centre to the point where circle rouches x-axis should be 3. it is going out of circle.

The key to cracking as always is figuring out all the current possiblities and checking whether we are able to get to the requisite condition
State 1 provides :
Coordinates at a distance of 2 units from (3,4) lie on the circumference of a circle with center (3,4) and radius 2 units. Thus all the points on the circumference of this circle can be A's coordinates. Thus insufficient
State 2 provides :
Coordinates at a distance of 3 units from (0,0) lie on the circumference of a circle with center (0,0) and radius 3 units. Thus all the points on the circumference of this circle can be A's coordinates. Thus this statement is not sufficient.

When combined together we get a unique point of intersection thus clearly sufficient when the 2 equations are taken into consideration
Hence IMO C

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