JAMIA 2022 — Mathematics PYQ

Step 1: Simplify the target expression

We want to find the minimum value of $|2z-6+5i|$. Let's factor out a $2$:

$|2z-6+5i|=|2(z-3+\frac{5}{2}i)|=2|z-(3-\frac{5}{2}i)|$

Let $z_1=3+2i$ and $z_2=3-\frac{5}{2}i$.

The given condition is $|z-z_1|\le 2$ (representing a disk centered at $z_1$ with radius $r=2$).

We need to find the minimum value of $2|z-z_2|$.

Step 2: Geometric Interpretation

• The condition $|z-(3+2i)|\le 2$ represents a circle (and its interior) with center $C(3,2)$ and radius $r=2$.

• The expression $|z-(3-\frac{5}{2}i)|$ represents the distance between a point $z$ in the disk and the point $P(3,-2.5)$.

Step 3: Calculate the distance between the center and the point

Let $d$ be the distance between the center of the circle $C(3,2)$ and the point $P(3,-2.5)$:

$d=\sqrt{(3-3{)}^2+(2-(-2.5){)}^2}$

$d=\sqrt{0^2+(4.5{)}^2}=4.5=\frac{9}{2}$

Step 4: Find the minimum distance $|z-z_2|$

The minimum distance from a point $P$ outside a circle to any point $z$ on/inside the circle is:

$|z-z_2{|}_{min}=d-r$

$|z-z_2{|}_{min}=\frac{9}{2}-2=\frac{5}{2}$

Step 5: Final Calculation

The original expression we needed to evaluate was $2|z-z_2|$.

Minimum value $=2\times \frac{5}{2}=5$

Final Answer:

The minimum value is $5$.