CUET UG 2025 — Mathematics Calculus

The abscissa (x-coordinate) and ordinate (y-coordinate) change at the same rate means:

$$\frac{dx}{dt} = \frac{dy}{dt}$$dtdx​=dtdy​

where $t$t represents time.

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Given the curve $y^2 = 8x$y2=8x, differentiate both sides with respect to time $t$t:

$$2y \cdot \frac{dy}{dt} = 8 \cdot \frac{dx}{dt}$$2y⋅dtdy​=8⋅dtdx​

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Since $\frac{dx}{dt} = \frac{dy}{dt}$dtdx​=dtdy​, substitute:

$$2y \cdot \frac{dx}{dt} = 8 \cdot \frac{dx}{dt}$$2y⋅dtdx​=8⋅dtdx​

Dividing both sides by $\frac{dx}{dt}$dtdx​:

$$2y = 8$$2y=8

$$y = 4$$y=4

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Substitute $y = 4$y=4 into the curve equation $y^2 = 8x$y2=8x:

$$(4)^2 = 8x$$(4)2=8x

$$16 = 8x$$16=8x

$$x = 2$$x=2

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Therefore, the point is $(2, 4)$(2,4).