JEE 2023 Mathematics PYQ — A triangle is formed by the tangents at the point on the curves a… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
A triangle is formed by the tangents at the point (2,2) on the curves y2=2x and x2+y2=4x, and the line x+y+2=0. If r is the radius of its circumcircle, then r2 is equal to:
Choose the correct answer:
A.
10
(Correct Answer)
B.
9
C.
8
D.
7
Correct Answer:
10
Explanation
Solution
Step 1: Find the equation of Tangent L1 to the curve y2=2x at (2,2)
Using the formula yy1=a(x+x1) for y2=4ax (yahan 4a=2, isliye a=1/2):
2y=1(x+2)
x−2y+2=0…(L1)
Step 2: Find the equation of Tangent L2 to the curve x2+y2=4x at (2,2)
Using xx1+yy1=2(x+x1):
2x+2y=2(x+2)
2x+2y=2x+4
2y=4⟹y=2…(L2)
Step 3: The third line is given as L3
x+y+2=0…(L3)
Step 4: Find the vertices of the triangle
Vertex A (L1∩L2): Put y=2 in x−2y+2=0.
x−4+2=0⟹x=2. So, A=(2,2).
Vertex B (L2∩L3): Put y=2 in x+y+2=0.
x+2+2=0⟹x=−4. So, B=(−4,2).
Vertex C (L1∩L3): Solve x−2y=−2 and x+y=−2.
Subtracting gives −3y=0⟹y=0. Then x=−2. So, C=(−2,0).
Step 5: Calculate side lengths of △ABC
Using distance formula:
AB=(2−(−4))2+(2−2)2=62=6
BC=(−4−(−2))2+(2−0)2=(−2)2+22=8
AC=(2−(−2))2+(2−0)2=42+22=20
Step 6: Calculate Area (Δ) and Circumradius (r)
Area of triangle with coordinates (2,2),(−4,2),(−2,0):
Δ=21∣2(2−0)+(−4)(0−2)+(−2)(2−2)∣
Δ=21∣4+8+0∣=6
Using the formula r=4Δabc:
r=4×66×8×20=4160
r2=16160=10
Final Answer:
r2=10
Explanation
Solution
Step 1: Find the equation of Tangent L1 to the curve y2=2x at (2,2)
Using the formula yy1=a(x+x1) for y2=4ax (yahan 4a=2, isliye a=1/2):
2y=1(x+2)
x−2y+2=0…(L1)
Step 2: Find the equation of Tangent L2 to the curve x2+y2=4x at (2,2)
Using xx1+yy1=2(x+x1):
2x+2y=2(x+2)
2x+2y=2x+4
2y=4⟹y=2…(L2)
Step 3: The third line is given as L3
x+y+2=0…(L3)
Step 4: Find the vertices of the triangle
Vertex A (L1∩L2): Put y=2 in x−2y+2=0.
x−4+2=0⟹x=2. So, A=(2,2).
Vertex B (L2∩L3): Put y=2 in x+y+2=0.
x+2+2=0⟹x=−4. So, B=(−4,2).
Vertex C (L1∩L3): Solve x−2y=−2 and x+y=−2.
Subtracting gives −3y=0⟹y=0. Then x=−2. So, C=(−2,0).
Step 5: Calculate side lengths of △ABC
Using distance formula:
AB=(2−(−4))2+(2−2)2=62=6
BC=(−4−(−2))2+(2−0)2=(−2)2+22=8
AC=(2−(−2))2+(2−0)2=42+22=20
Step 6: Calculate Area (Δ) and Circumradius (r)
Area of triangle with coordinates (2,2),(−4,2),(−2,0):
