NIMCET 2023 — Mathematics PYQ

Step 1: Parabola Properties and Focal Distance Formula

For a standard parabola given by the equation:

$$y^2=4ax$$

• The coordinates of its focus $S$ are $(a,0)$.

• The equation of its directrix is $x=-a$.

By definition, the focal distance of any parametric point $P(x,y)$ lying on the parabola is given by the formula:

$$\text{Focal Distance }(PF)=x+a$$

Step 2: Calculate the Coordinates of Point $P$

We are given that the focal distance is $5a$. Let's set up the equation to find the $x$-coordinate of point $P$:

$$x+a=5a$$

$$x=5a-a$$

$$x=4a$$

Now, substitute $x=4a$ back into the original parabola equation $y^2=4ax$ to find the corresponding $y$-coordinate:

$$y^2=4a(4a)$$

$$y^2=16a^2$$

$$y=\pm \sqrt{16a^2}=\pm 4a$$

Since the problem explicitly specifies that point $P$ lies in the first quadrant, both its coordinates must be positive. Therefore:

$$y=4a$$

Thus, the exact position of point $P$ is:

$$P=(4a,4a)$$

Step 3: Determine the Locus of Point $P$

Since $P$ is a specific point determined by the fixed distance condition, we observe the relationship between its coordinates:

• $x=4a$

• $y=4a$

This means that for any positive parameter value $a$, the coordinates satisfy the path condition:

$$y=x⟹x-y=0$$

Step 4: Verify the Given Options

Let's check which of the given options satisfies the locus condition $y=x$:

• A) $(1,0)⟹1\ne 0$ (Incorrect)

• B) $(1,1)⟹1=1$ (Correct)

• C) $(0,2)⟹0\ne 2$ (Incorrect)

• D) $(2,0)⟹2\ne 0$ (Incorrect)

Conclusion

The point $(1,1)$ satisfies the locus condition.

The correct option is B) (1,1).