Equal rate condition

Find the point on a curve where the x-coordinate and y-coordinate are changing at the same rate.

2 questions · Standard +0.0

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CAIE P1 2023 March Q3
4 marks Moderate -0.3
3 A curve has equation \(y = \frac { 1 } { 60 } ( 3 x + 1 ) ^ { 2 }\) and a point is moving along the curve.
Find the \(x\)-coordinate of the point on the curve at which the \(x\) - and \(y\)-coordinates are increasing at the same rate.
CAIE P1 2020 Specimen Q8
6 marks Standard +0.3
8 A curve has equation \(y = \frac { 12 } { 3 - 2 x }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.
  2. Find the possible \(x\)-coordinates of \(A\).