JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023Let z1=2+3i and z2=3+4i. The set S={z∈C:∣z−z1∣2−∣z−z2∣2=c} represents a:
Choose the correct answer:
- A.
hyperbola with the length of the transverse axis 7
- B.
hyperbola with eccentricity 2
- C.
straight line with the sum of its intercepts on the coordinate axes equals −18
- D.
straight line with the sum of its intercepts on the coordinate axes equals 14
(Correct Answer)
straight line with the sum of its intercepts on the coordinate axes equals 14
Explanation
Given: z1=2+3i,z2=3+4i
And S={z∈C:∣z−z1∣2−∣z−z2∣2=∣z1−z2∣2}
Let z=x+iy
Now, (z−z1)=(x−2)+i(y−3)
(z−z2)=(x−3)+i(y−4)
(z1−z2)=−1−i
∴∣z−z1∣2−∣z−z2∣2=∣z1−z2∣2
⇒[(x−2)2+(y−3)2]−[(x−3)2+(y−4)2]=1+1
⇒x+y=7
⇒7x+7y=1
Final Answer :
So, S represents a straight line with the sum of its intercepts on coordinate axis equals 14.
Explanation
Given: z1=2+3i,z2=3+4i
And S={z∈C:∣z−z1∣2−∣z−z2∣2=∣z1−z2∣2}
Let z=x+iy
Now, (z−z1)=(x−2)+i(y−3)
(z−z2)=(x−3)+i(y−4)
(z1−z2)=−1−i
∴∣z−z1∣2−∣z−z2∣2=∣z1−z2∣2
⇒[(x−2)2+(y−3)2]−[(x−3)2+(y−4)2]=1+1
⇒x+y=7
⇒7x+7y=1
Final Answer :
So, S represents a straight line with the sum of its intercepts on coordinate axis equals 14.