A delivery drone icon for a freezer-label icon, tracked as part of a corner shop, is translated twice on a coordinate grid. Point \(A\) starts at \((2, 1)\)...
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A delivery drone icon for a freezer-label icon, tracked as part of a corner shop, is translated twice on a coordinate grid. Point \(A\) starts at \((2, 1)\) and is translated by the vector \(\binom{x}{3}\) to reach the point \((6, 4)\).
- Find the value of \(x\). (2)
- The image point is then translated by the vector \(\binom{-1}{y}\) to reach the point \((5, 10)\). Find the value of \(y\). (2)
Answer Details
(a) A translation by the vector \(\binom{x}{3}\) adds \(x\) to the starting \(x\)-coordinate and adds 3 to the starting \(y\)-coordinate. Point \(A\) starts at \((2, 1)\) and reaches \((6, 4)\), so looking only at the \(x\)-coordinates, \(2 + x = 6\), which gives \(x = 4\) [2 marks]. This can be checked against the \(y\)-coordinates too: \(1 + 3 = 4\), which matches the given image point.
(b) The image point after the first translation is \((6, 4)\). Applying the second translation \(\binom{-1}{y}\) takes this to \((5, 10)\), so looking at the \(y\)-coordinates, \(4 + y = 10\), which gives \(y = 6\) [2 marks]. Checking the \(x\)-coordinates: \(6 + (-1) = 5\), which matches the final point given.
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