JEE 2023 Mathematics PYQ — Let be the circle in the complex plane with centre and radius . L… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let C be the circle in the complex plane with centre z0=21(1+3i) and radius r=1. Let z1=1+i and the complex number z2 be outside the circle C such that ∣z1−z0∣∣z2−z0∣=1. If z0,z1 and z2 are collinear, then the smaller value of ∣z2∣2 is equal to
Choose the correct answer:
A.
27
B.
213
C.
25
(Correct Answer)
D.
23
Correct Answer:
25
Explanation
Step 1: Given Data ko Samajhna
Centre:z0=21+23i
Radius:r=1
Point 1:z1=1+i
Condition:∣z1−z0∣⋅∣z2−z0∣=1
Collinearity:z0,z1, aur z2 ek hi seedhi rekha (line) par hain.
Step 2: Distance ∣z1−z0∣ Nikalna
z1−z0=(1+i)−(21+23i)=21−21i
Iska magnitude (distance) hoga:
∣z1−z0∣=(21)2+(−21)2=41+41=21=21
Step 3: Distance ∣z2−z0∣ Nikalna
Di gayi condition se:
21⋅∣z2−z0∣=1⟹∣z2−z0∣=2
Step 4: Collinearity ka Use Karke z2 Find Karna
Chunki z0,z1,z2 collinear hain, z2 ko hum vector form mein aise likh sakte hain:
z2=z0+λ(z1−z0)
Yahan λ ek real number hai. Magnitude check karte hain:
∣z2−z0∣=∣λ∣⋅∣z1−z0∣
2=∣λ∣⋅21⟹∣λ∣=2⟹λ=±2
Ab hamare paas z2 ki do possible values hain:
Case 1 (λ=2):z2=z0+2(z1−z0)=2z1−z0
z2=2(1+i)−(21+23i)=2+2i−21−23i=23+21i
Case 2 (λ=−2):z2=z0−2(z1−z0)=3z0−2z1
z2=3(21+23i)−2(1+i)=23+29i−2−2i=−21+25i
Step 5: ∣z2∣2 ki Smaller Value Nikalna
Hame ∣z2∣2 ki smaller value chahiye:
Case 1:∣z2∣2=(23)2+(21)2=49+41=410=25
Case 2:∣z2∣2=(−21)2+(25)2=41+425=426=213
Dono mein se choti value 25 hai.
Sahi Answer: (3) 25
Explanation
Step 1: Given Data ko Samajhna
Centre:z0=21+23i
Radius:r=1
Point 1:z1=1+i
Condition:∣z1−z0∣⋅∣z2−z0∣=1
Collinearity:z0,z1, aur z2 ek hi seedhi rekha (line) par hain.
Step 2: Distance ∣z1−z0∣ Nikalna
z1−z0=(1+i)−(21+23i)=21−21i
Iska magnitude (distance) hoga:
∣z1−z0∣=(21)2+(−21)2=41+41=21=21
Step 3: Distance ∣z2−z0∣ Nikalna
Di gayi condition se:
21⋅∣z2−z0∣=1⟹∣z2−z0∣=2
Step 4: Collinearity ka Use Karke z2 Find Karna
Chunki z0,z1,z2 collinear hain, z2 ko hum vector form mein aise likh sakte hain:
z2=z0+λ(z1−z0)
Yahan λ ek real number hai. Magnitude check karte hain:
