If \(2x^2-7xy+3y^2=0\), then find the ratio \(x:y\).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Both 1 and 2
The expression is a quadratic in x and y, so it is factorised by splitting the middle term: \(-7xy=-6xy-xy\). A product is zero when either factor is zero, which gives two ratios.
\(2x^2-6xy-xy+3y^2=2x(x-3y)-y(x-3y)=(2x-y)(x-3y)\)
\(2x-y=0\) gives \(2x=y\), so \(x:y=1:2\)
\(x-3y=0\) gives \(x=3y\), so \(x:y=3:1\)
Both ratios satisfy the equation, so Options (1) and (2) are both correct, which is what Option (3) says.