JEE 2023 Mathematics PYQ — Let and be two points on the line such that and are symmetric wit… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let B and C be two points on the line y+x=0 such that B and C are symmetric with respect to the origin. Suppose A is a point on y−2x=2 such that ΔABC is an equilateral triangle. Then, the area of the ΔABC is:
Choose the correct answer:
A.
310
B.
33
C.
23
D.
38
(Correct Answer)
Correct Answer:
38
Explanation
Solution
Step 1: Identify the properties of the triangle
Points B and C lie on the line L1:x+y=0.
Since B and C are symmetric with respect to the origin (0,0), the origin is the midpoint of the side BC.
In an equilateral triangle, the altitude from vertex A to the opposite side BC passes through the midpoint of BC. Thus, the altitude of ΔABC is the distance from point A to the origin.
Step 2: Determine the line of the altitude
The altitude is perpendicular to L1:x+y=0.
The slope of L1 is −1. Therefore, the slope of the altitude is 1.
Since the altitude passes through the origin, its equation is y=x.
Step 3: Find the coordinates of vertex A
Point A is the intersection of the altitude y=x and the given line L2:y−2x=2.
Substitute y=x into y−2x=2:
x−2x=2⟹−x=2⟹x=−2
Since y=x, the coordinates of A are (−2,−2).
Step 4: Calculate the length of the altitude (h)
The height h is the distance from A(−2,−2) to the origin (0,0):
h=(−2−0)2+(−2−0)2=4+4=8=22
Step 5: Calculate the area of the equilateral triangle
The relationship between the height (h) and the side (a) of an equilateral triangle is h=23a, which implies a=32h.
The formula for the area is:
Area=43a2=43(32h)2=43⋅34h2=3h2
Substitute h=22 into the area formula:
Area=3(22)2=38
Final Answer
The area of ΔABC is 38. Therefore, option (4) is correct.
Explanation
Solution
Step 1: Identify the properties of the triangle
Points B and C lie on the line L1:x+y=0.
Since B and C are symmetric with respect to the origin (0,0), the origin is the midpoint of the side BC.
In an equilateral triangle, the altitude from vertex A to the opposite side BC passes through the midpoint of BC. Thus, the altitude of ΔABC is the distance from point A to the origin.
Step 2: Determine the line of the altitude
The altitude is perpendicular to L1:x+y=0.
The slope of L1 is −1. Therefore, the slope of the altitude is 1.
Since the altitude passes through the origin, its equation is y=x.
Step 3: Find the coordinates of vertex A
Point A is the intersection of the altitude y=x and the given line L2:y−2x=2.
Substitute y=x into y−2x=2:
x−2x=2⟹−x=2⟹x=−2
Since y=x, the coordinates of A are (−2,−2).
Step 4: Calculate the length of the altitude (h)
The height h is the distance from A(−2,−2) to the origin (0,0):
h=(−2−0)2+(−2−0)2=4+4=8=22
Step 5: Calculate the area of the equilateral triangle
The relationship between the height (h) and the side (a) of an equilateral triangle is h=23a, which implies a=32h.
The formula for the area is:
Area=43a2=43(32h)2=43⋅34h2=3h2
Substitute h=22 into the area formula:
Area=3(22)2=38
Final Answer
The area of ΔABC is 38. Therefore, option (4) is correct.
