Maths Q2 – JEE-MAIN 2026

Home Maths Complex & QuadraticQuestion #70

Let the circles

C_{1} : | z | = r

and

C_{2} : | z - 3 - 4 i | = 5

z \in C

, be such that

C_{2}

lies within

C_{1}

. If

z_{1}

moves on

C_{1}

z_{2}

moves on

C_{2}

and

min | z_{1} - z_{2} | = 2

, then

max | z_{1} - z_{2} |

is equal to :

Clue 1 of 3: Geometric Interpretation of Circles

Understand the properties of circles in the complex plane, specifically their centers and radii, and how distances between points on them are related to the distance between their centers.

🧠 Full Solution Path

Step 1: Identify Circle Parameters and Distance between Centers✦ Active

Circle $C_{1} : | z | = r$ . Its center is $O_{1} = 0$ and its radius is $R_{1} = r$ .

Circle $C_{2} : | z - (3 + 4 i) | = 5$ . Its center is $O_{2} = 3 + 4 i$ and its radius is $R_{2} = 5$ .

The distance between the centers $O_{1}$ and $O_{2}$ is $d = | O_{1} - O_{2} | = | 0 - (3 + 4 i) | = | - 3 - 4 i |$ .

d = \sqrt{(- 3)^{2} + (- 4)^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5.

Step 2: Use Minimum Distance to Find Radius

r

○ Expand

Since $C_{2}$ lies within $C_{1}$ , the minimum distance between a point on $C_{1}$ and a point on $C_{2}$ is given by $R_{1} - d - R_{2}$ .

We are given that $min | z_{1} - z_{2} | = 2$ . Therefore, we can set up the equation:

r - d - R_{2} = 2

Substitute the known values $d = 5$ and $R_{2} = 5$ :

r - 5 - 5 = 2

Solving for $r$ :

r - 10 = 2 ⟹ r = 12.

This value of $r = 12$ satisfies the condition for $C_{2}$ to be within $C_{1}$ , which is $d + R_{2} < R_{1} ⟹ 5 + 5 < 12 ⟹ 10 < 12$ .

💡 Teacher's Secret Hint

Ensure the condition for one circle lying within another ( $d + R_{2} < R_{1}$ ) is met after calculating $r$ .

Step 3: Calculate Maximum Distance○ Expand

The maximum distance between a point on $C_{1}$ and a point on $C_{2}$ is given by $R_{1} + d + R_{2}$ .

Substitute the values $r = 12$ , $d = 5$ , and $R_{2} = 5$ :

max | z_{1} - z_{2} | = r + d + R_{2} = 12 + 5 + 5 = 22.

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Maths Question 2 – JEE-MAIN 2026

Clue 1 of 3: Geometric Interpretation of Circles

🧠 Full Solution Path