Discussion
If \(6x^2 - 4xy - 10y^2 = 0\) then what is \(x : y\)?
Answer: B
Divide the equation by 2 and factorise it as a quadratic in x with y as the parameter.
\[ 3x^2 - 2xy - 5y^2 = 0 \Rightarrow (3x - 5y)(x + y) = 0 \]
The first factor gives \(3x = 5y\), so \(x : y = 5 : 3\). The second factor gives \(x = -y\), so \(x : y = -1 : 1\). So the ratio is \((-1 : 1)\) or \((5 : 3)\). The options with 3 : 5 have the ratio the wrong way round, and the ones with \(-2 : 1\) do not satisfy the equation (for \(x = -2y\) the left side is \(12 + 8 - 10 = 10\) times \(y^2\), which is not zero). The answer is option B.
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