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Distinct points P, Q, R, S lie on the circle x^2 + y^2 = 25 and have

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Bunuel
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Distinct points P, Q, R, S lie on the circle x^2 + y^2 = 25 and have [#permalink] New post   10 May 2019, 00:48
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Distinct points P, Q, R, S lie on the circle x^2 + y^2 = 25 and have integer coordinates. The distances PQ and RS are irrational numbers. What is the greatest possible value of the ratio \(\frac{PQ}{RS}\)?

(A) 3

(B) 5

(C) \(3\sqrt{5}\)

(D) 7

(E) \(5\sqrt{2}\)
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Re: Distinct points P, Q, R, S lie on the circle x^2 + y^2 = 25 and have [#permalink] New post   10 May 2019, 01:17
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Bunuel wrote:
Distinct points P, Q, R, S lie on the circle x^2 + y^2 = 25 and have integer coordinates. The distances PQ and RS are irrational numbers. What is the greatest possible value of the ratio \(\frac{PQ}{RS}\)?

(A) 3

(B) 5

(C) \(3\sqrt{5}\)

(D) 7

(E) \(5\sqrt{2}\)



    • We know that the points P, Q, R, and R have integers coordinates.
    • Since they lie on the on the circle \(x^2 + y^2 = 25\), they will satisfy this equation.
      o Hence, we need to think of all the possible ways in which x and y can be integers and give us the value 25.

\(x^2 + y^2 = 25\) is possible in the following cases:
    1. |x| = 3 and |y| = 4
    2. |x| = 4 and |y| = 3
    3. |x| = 0 and |y| = 5
    4. |x| = 5 and |y| = 0

The reason for using modulus sign is because x and y can be negative or positive both so we need to consider all the cases.

Now, we know that PQ and RS are irrational numbers, so we need to look for a combination of the above cases, in which we can get PQ and RS as irrational numbers.

Also, since we want the maximum value of PQ/RS, we need to ensure that PQ is a great as possible and RS is as small as possible.

    • The distance formula is \(\sqrt((x_2 – x_1)^2 + (y_2 – y_1)^2)\)
      o To find the maximum value of PQ, we need to look for opposite sign of \(x_2\) and \(x_1\), so that when we subtract, we get maximum value and the same goes for \(y_2\) and \(y_1\)
         For example, P can be (3, 4) and Q can be (-4, -3)
      o This way on adding up, we’ll get
         \(\sqrt((3 – (-4))^2 + (4– (-3))^2)\) \(= \sqrt(98) = 7*\sqrt(2)\)
      o To find the minimum value, we need to take values, which are very close to each other.
         For example R can be (3, -4) and S can be (4, -3)
         The distance in this case will be:
         \(\sqrt((3 – (4))^2 + (-4– (-3))^2)\) = \(\sqrt(2)\)

Thus, the ratio of PQ/RS will be \(7\sqrt(2)/\sqrt(2) = 7\)

Hence the answer should be Option D.
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Re: Distinct points P, Q, R, S lie on the circle x^2 + y^2 = 25 and have [#permalink] New post   11 May 2019, 00:03
Points P,Q,R and S have integer coordinates and they lie on a circle of radius 5 and origin as center .

Possible coordinates for point P,Q,R and S are ={(5,0),(4,3,),(3,4),(0,5),(-3,4),(-4,3),(-5,0),(-4,-3),(-3,-4),(0,-5),(3,-4),and (4,-3)}

For PQ/RS to be max ...PQ has to be max and RS has to be minimum and both distances are to be irrational values

Since Distance PQ and RS are to be irrational ..PQ and RS can not be diagonal as in that case value =10 will be rational.
Lets check next possible distance but both diagnol ..say points (4,3) and (-3,-4) distance is 7Sq Root(2) a irrational value ...

Now for RS let us take two immediate points say (-3,4) and (-4,3) distance is sqrot(2)

Ratio of PQ/RS is =7...hence answer is D
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Re: Distinct points P, Q, R, S lie on the circle x^2 + y^2 = 25 and have [#permalink] New post   11 May 2019, 00:36
I think we know the max value can be near to 10 since we can not take 10 which is integer. 10 is Square root of 100 we can think of square root of 99 and 98 to test the max irrational value.

Min value can be 1 since we can not take 0. 1 is integer so we can take min value as sqrt of 2 ( since we get irrational numbers from sqrt of integers which are not perfect square)

If we test sqrt of 98 / sqrt of 2 then we get 7 as answer.

Hope my logic is correct.

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Re: Distinct points P, Q, R, S lie on the circle x^2 + y^2 = 25 and have [#permalink]
11 May 2019, 00:36
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