NIMCET 2007 — Mathematics PYQ

Step 1: Understand the given conditions

Let the required point on the parabola be $(x,y)$.

• The term abscissa refers to the $x$-coordinate, and its rate of change with respect to time $t$ is $\frac{dx}{dt}$.

• The term ordinate refers to the $y$-coordinate, and its rate of change with respect to time $t$ is $\frac{dy}{dt}$.

According to the given problem statement, the ordinate increases at twice the rate of the abscissa:

$$\frac{dy}{dt}=2\cdot \frac{dx}{dt}$$

--- (Equation 1)

Step 2: Differentiate the equation of the parabola

The given equation of the parabola is:

$$y^2=18x$$

Differentiating both sides with respect to time $t$ using the chain rule:

$$2y\cdot \frac{dy}{dt}=18\cdot \frac{dx}{dt}$$

Simplify the equation by dividing both sides by 2:

$$y\cdot \frac{dy}{dt}=9\cdot \frac{dx}{dt}$$

--- (Equation 2)

Step 3: Find the $y$-coordinate

Substitute the value of $\frac{dy}{dt}$ from Equation 1 into Equation 2:

$$y\cdot \left(2\cdot \frac{dx}{dt}\right)=9\cdot \frac{dx}{dt}$$

Since the rate of change $\frac{dx}{dt}\ne 0$, we can cancel it from both sides:

$$2y=9$$

$$y=\frac{9}{2}$$

Step 4: Find the $x$-coordinate

Substitute the value of $y=\frac{9}{2}$ back into the original equation of the parabola ($y^2=18x$):

$${\left(\frac{9}{2}\right)}^2=18x$$

$$\frac{81}{4}=18x$$

$$x=\frac{81}{4\times 18}$$

Divide the numerator and denominator by 9:

$$x=\frac{9}{4\times 2}$$

$$x=\frac{9}{8}$$

Conclusion

The required point on the parabola is $(\frac{9}{8},\frac{9}{2})$.

Therefore, the correct option is (d).