Let z$z$ be the complex number satisfying |z-5|leq 3$|z-5|\leq 3$ and having maximum positive principal argument. Then 34left|frac5z-125iz+16right|^2$34\left|\frac{5z-12}{5iz+16}\right|^{2}$ is equal to:

### Related Formula textFor maximum argument of z text on a circle, the ray from origin is tangent to the circle.$$\text{For maximum argument of } z \text{ on a circle, the ray from origin is tangent to the circle.}$$ textIf |z-a| leq r, text maximum argument implies sintheta = fracr|a| text and coordinates are x=acos^2theta, y=asinthetacostheta$$\text{If } |z-a| \leq r, \text{ maximum argument implies } \sin\theta = \frac{r}{|a|} \text{ and coordinates are } x=a\cos^2\theta, y=a\sin\theta\cos\theta$$ ### Core Logic

The condition |z-5|leq 3$|z-5|\leq 3$ represents a disk centered at (5,0)$(5,0)$ with radius 3$3$. To maximize the principal argument theta$\theta$, the ray from origin must touch the circle in the first quadrant. The tangent, origin, and center form a right-angled triangle. ### Step 1: Locate the Point z From geometry, hypotenuse c = 5$c = 5$, opposite (radius) r = 3$r = 3$. The adjacent side (length of tangent) is sqrt5^2 - 3^2 = 4$\sqrt{5^2 - 3^2} = 4$. The angle of tangency theta$\theta$ satisfies sintheta = frac35$\sin\theta = \frac{3}{5}$ and costheta = frac45$\cos\theta = \frac{4}{5}$. The point P(z)$P(z)$ lies on the circle and the tangent ray: z equiv (4costheta, 4sintheta) = left(4left(frac45right), 4left(frac35right)right) = left(frac165, frac125right)$$z \equiv (4\cos\theta, 4\sin\theta) = \left(4\left(\frac{4}{5}\right), 4\left(\frac{3}{5}\right)\right) = \left(\frac{16}{5}, \frac{12}{5}\right)$$ z = frac165 + frac125i$$z = \frac{16}{5} + \frac{12}{5}i$$ ### Step 2: Evaluate the Target Expression We need 34left|frac5z - 125iz + 16right|^2$34\left|\frac{5z - 12}{5iz + 16}\right|^2$. Substitute 5z = 16 + 12i$5z = 16 + 12i$: Numerator: 5z - 12 = 16 + 12i - 12 = 4 + 12i$5z - 12 = 16 + 12i - 12 = 4 + 12i$ Denominator: 5iz + 16 = i(16 + 12i) + 16 = 16i - 12 + 16 = 4 + 16i$5iz + 16 = i(16 + 12i) + 16 = 16i - 12 + 16 = 4 + 16i$ Expression: 34 left|frac4 + 12i4 + 16iright|^2 = 34 frac|4 + 12i|^2|4 + 16i|^2 = 34 frac4^2 + 12^24^2 + 16^2$$34 \left|\frac{4 + 12i}{4 + 16i}\right|^2 = 34 \frac{|4 + 12i|^2}{|4 + 16i|^2} = 34 \frac{4^2 + 12^2}{4^2 + 16^2}$$ = 34 left(frac16 + 14416 + 256right) = 34 left(frac160272right)$$= 34 \left(\frac{16 + 144}{16 + 256}\right) = 34 \left(\frac{160}{272}\right)$$ = frac34 times 160272 = frac5440272 = 20$$= \frac{34 \times 160}{272} = \frac{5440}{272} = 20$$ ### Pattern Recognition For maximum arg(z)$\arg(z)$ on |z-c| = r$|z-c| = r$ where c$c$ is real, z$z$ coordinates are given by geometric projection: z = sqrtc^2-r^2(costheta + isintheta)$z = \sqrt{c^2-r^2}(\cos\theta + i\sin\theta)$ where sintheta = r/c$\sin\theta = r/c$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Complex Numbers