NIMCET 2025 Mathematics PYQ — An equilateral triangle is inscribed in the parabola . One vertex… | Mathem Solvex | Mathem Solvex
Tip:A–D to answerE for explanationV for videoS to reveal answer
NIMCET 2025 — Mathematics PYQ
NIMCET | Mathematics | 2025
An equilateral triangle is inscribed in the parabola y2=x. One vertex of the triangle is at the vertex of the parabola. The centroid of the triangle is:
Choose the correct answer:
A.
(2, 0)
(Correct Answer)
B.
(2,0)
C.
(1, 0)
D.
(3,0)
Correct Answer:
(2, 0)
Explanation
Step 1: Understand the Geometry and Assume Coordinates
The equation of the parabola is given as:
y2=x
The vertex of this parabola is at the origin, O(0,0). Let this be the first vertex of our inscribed equilateral triangle OAB.
Since a parabola is completely symmetric about its axis (here, the x-axis where y=0), an equilateral triangle with one vertex at the origin must also be symmetric about the x-axis.
Let the coordinates of vertex A be (t2,t).
By symmetry, the coordinates of vertex B must be (t2,−t).
Step 2: Apply the Condition for an Equilateral Triangle
In an equilateral triangle, all three sides are equal in length, so OA=AB.
Using the distance formula, let's find the square of the lengths OA2 and AB2:
Calculate OA2 (Distance from origin (0,0) to A(t2,t)):
OA2=(t2−0)2+(t−0)2
OA2=t4+t2
Calculate AB2 (Distance between A(t2,t) and B(t2,−t)):
AB2=(t2−t2)2+(t−(−t))2
AB2=02+(2t)2
AB2=4t2
Equating the two values (OA2=AB2):
t4+t2=4t2
t4−3t2=0
t2(t2−3)=0
Since t=0 (otherwise all vertices collapse to the origin), we solve for t2:
t2=3⟹t=3 or t=−3
Substituting t2=3 back into our vertex assumptions gives the explicit coordinates of the points:
O=(0,0)
A=(3,3)
B=(3,−3)
Step 3: Calculate the Centroid of △OAB
The formula for the centroid (G) of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) is:
G=(3x1+x2+x3,3y1+y2+y3)
Substitute the coordinates of O, A, and B:
G=(30+3+3,30+3+(−3))
G=(36,30)
G=(2,0)
Correct Answer
Correct Option: (D)
Explanation
Step 1: Understand the Geometry and Assume Coordinates
The equation of the parabola is given as:
y2=x
The vertex of this parabola is at the origin, O(0,0). Let this be the first vertex of our inscribed equilateral triangle OAB.
Since a parabola is completely symmetric about its axis (here, the x-axis where y=0), an equilateral triangle with one vertex at the origin must also be symmetric about the x-axis.
Let the coordinates of vertex A be (t2,t).
By symmetry, the coordinates of vertex B must be (t2,−t).
Step 2: Apply the Condition for an Equilateral Triangle
In an equilateral triangle, all three sides are equal in length, so OA=AB.
Using the distance formula, let's find the square of the lengths OA2 and AB2:
Calculate OA2 (Distance from origin (0,0) to A(t2,t)):
OA2=(t2−0)2+(t−0)2
OA2=t4+t2
Calculate AB2 (Distance between A(t2,t) and B(t2,−t)):
AB2=(t2−t2)2+(t−(−t))2
AB2=02+(2t)2
AB2=4t2
Equating the two values (OA2=AB2):
t4+t2=4t2
t4−3t2=0
t2(t2−3)=0
Since t=0 (otherwise all vertices collapse to the origin), we solve for t2:
t2=3⟹t=3 or t=−3
Substituting t2=3 back into our vertex assumptions gives the explicit coordinates of the points:
O=(0,0)
A=(3,3)
B=(3,−3)
Step 3: Calculate the Centroid of △OAB
The formula for the centroid (G) of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) is:
