question_answer

Find the point on the curve \[{{y}^{2}}=8x\] for which for the abscissa and ordinate change at the same rate.                     

Answer:

Let the required point be (x, y).                         Given,   \[\frac{dy}{dt}=\frac{dx}{dt}\] and       \[{{y}^{2}}=8x\]            On differentiating both sides of Eq. (ii) w.r.t.t, (we get \[2y\frac{dy}{dt}=8\cdot \frac{dx}{dt}\] \[\Rightarrow \] \[2y=8\]     [from Eq. (i)] \[\Rightarrow \]   \[y=4\]  On putting y = 4 in Eq. (ii), we get   \[8x=16\,\,\,\Rightarrow \,\,\,x=2\]         Hence, the required point is (2, 4).