NIMCET 2023 — Mathematics PYQ

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Question

NIMCET 2023 — Mathematics PYQ

NIMCET | Mathematics | 2023

A point P in the first quadrant, lies on $y^{2} = 4 a x$ , $a>0$ , and keep a distance of $5 a$ unit from its focus. Which of the following points lies on the locus of P?

Choose the correct answer:

Explanation

Step-by-step Derivation of the Locus

1. The focus of the parabola

y^{2} = 4 a x

is known to be

F (a, 0)

.

2. Let the point

P

on the parabola be represented by its parametric coordinates,

P (a t^{2}, 2 a t)

.

3. The distance between point

P

and the focus

F

is given by the distance formula:

P F = (a t^{2} - a)^{2} + (2 a t - 0)^{2}

.

4. It is given that this distance is

5 a

, so

P F = 5 a

.

5. Squaring both sides of the distance equation,

(5 a)^{2} = (a t^{2} - a)^{2} + (2 a t)^{2}

.

6. Expanding the terms,

25 a^{2} = a^{2} (t^{2} - 1)^{2} + 4 a^{2} t^{2}

.

7. Dividing by

a^{2}

(since

a &gt; 0

),

25 = (t^{2} - 1)^{2} + 4 t^{2}

.

8. Expanding the squared term,

25 = t^{4} - 2 t^{2} + 1 + 4 t^{2}

.

9. Simplifying the equation,

25 = t^{4} + 2 t^{2} + 1

.

10. This can be rewritten as

25 = (t^{2} + 1)^{2}

.

11. Taking the square root of both sides,

25 = (t^{2} + 1)^{2}

, which gives

5 = t^{2} + 1

.

12. Solving for

t^{2}

,

t^{2} = 5 - 1 = 4

.

13. Since

P

is in the first quadrant, both its

x

and

y

coordinates must be positive. This implies

t &gt; 0

, so

t = 4 = 2

.

14. Substituting

t = 2

back into the coordinates of

P

,

x = a (2^{2}) = 4 a

and

y = 2 a (2) = 4 a

.

15. Therefore, the point

P

is

(4 a, 4 a)

.

16. The locus of

P

is the set of all such points

(4 a, 4 a)

for any

a &gt; 0

.

17. The question asks which of the given points lies on this locus. This means checking which point has equal

x

and

y

coordinates.

18. Examining the options:

• (a)

(1, 0)

:

x \neq = y

.

• (b)

(1, 1)

:

x = y

. This point lies on the locus if

a = \frac{1}{4}

.

• (c)

(0, 2)

:

x \neq = y

.

• (d)

(2, 0)

:

x \neq = y

.

Final Answer

The point that lies on the locus of

P

is

(1, 1)

.

Choose the correct answer:

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