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Algebraic ratio application: Given (5x^2 − 3y^2) : (x y) = 11 : 2, determine the positive ratio x : y.

Difficulty: Medium

Correct Answer: 3 : 2

Explanation:


Introduction / Context:
Ratios involving quadratic expressions often reduce to a simple ratio between variables by substituting a single unknown for x/y. This question tests algebraic manipulation and solving a quadratic in a single variable that represents the ratio x : y.



Given Data / Assumptions:

  • (5x^2 − 3y^2) : (x y) = 11 : 2.
  • x and y are real; we are asked for the positive ratio x : y.
  • We assume y ≠ 0 so the ratio x/y is defined.


Concept / Approach:
Convert the proportion to an equation, then reduce the two-variable expression to a single variable by setting r = x / y. This yields a quadratic equation in r, from which we take the positive solution to obtain the ratio x : y.



Step-by-Step Solution:

(5x^2 − 3y^2) / (x y) = 11 / 2Cross-multiply: 2(5x^2 − 3y^2) = 11 x yLet r = x / y ⇒ x = r y. Substitute: 2(5r^2 y^2 − 3y^2) = 11 r y^2Divide by y^2 (y ≠ 0): 10 r^2 − 6 = 11 rRearrange: 10 r^2 − 11 r − 6 = 0Solve: Discriminant D = 121 + 240 = 361 ⇒ √D = 19r = (11 ± 19) / 20 ⇒ r = 30/20 = 3/2 or r = −8/20 = −2/5Positive ratio required ⇒ r = 3/2 ⇒ x : y = 3 : 2


Verification / Alternative check:
Pick y = 2, x = 3. Compute (5x^2 − 3y^2)/(xy) = (5*9 − 3*4)/(3*2) = (45 − 12)/6 = 33/6 = 11/2, matching the given proportion.



Why Other Options Are Wrong:
5 : 2, 7 : 2, 5 : 3, and 2 : 5 do not satisfy 10r^2 − 11r − 6 = 0 when r is formed as x/y.



Common Pitfalls:
Forgetting to divide through by y^2, sign errors when forming the quadratic, or selecting the negative root despite the requirement of a positive ratio.



Final Answer:
3 : 2

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