Circle Properties
We are given two circles in the complex plane:
- Circle $C_1$: $|z| = r$. This circle is centered at the origin $O=0$ with radius $R_1 = r$.
- Circle $C_2$: $|z - (3 + 4i)| = 5$. This circle is centered at $c_2 = 3 + 4i$ with radius $R_2 = 5$.
The problem states that $C_2$ lies entirely within $C_1$. This geometric condition implies $R_1 > d + R_2$, where $d$ is the distance between the centers.
Distance Between Centers
Calculate the distance $d$ between the centers of $C_1$ (origin $0$) and $C_2$ ($3+4i$):
$ d = |0 - (3 + 4i)| = |-(3 + 4i)| = \sqrt{(-3)^2 + (-4)^2} $
$ d = \sqrt{9 + 16} = \sqrt{25} = 5 $
Minimum Distance Calculation
When one circle ($C_2$) is inside another ($C_1$), the minimum distance between points $z_1$ on $C_1$ and $z_2$ on $C_2$ is given by the difference between the radii adjusted by the distance between centers:
$ \min|z_1 - z_2| = R_1 - (d + R_2) $
We are given $\min|z_1 - z_2| = 2$. Substitute the known values:
$ 2 = r - (5 + 5) $
$ 2 = r - 10 $
Solving for the radius $r$ of $C_1$:
$ r = 2 + 10 = 12 $
The radius of $C_1$ is $R_1 = 12$. Note that $12 > 5 + 5$, consistent with $C_2$ being inside $C_1$.
Maximum Distance Calculation
The maximum distance between points $z_1$ on $C_1$ and $z_2$ on $C_2$ occurs along the line connecting their centers, extending outwards:
$ \max|z_1 - z_2| = R_1 + d + R_2 $
Substitute the values $R_1 = 12$, $d = 5$, and $R_2 = 5$:
$ \max|z_1 - z_2| = 12 + 5 + 5 $
$ \max|z_1 - z_2| = 22 $
Similar Questions
Let A = {-3, -2, -1, 0, 1, 2, 3}. Let R be a relation on A defined by xRy if and only if $0\le x^2+2y\le 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $l + m$ is equal to
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Let $a \in \mathbf{R}$ and $A$ be a matrix of order $3 \times 3$ such that $\det (A) = -4$ and $A + I = \begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix}$, where $I$ is the identity matrix of order $3 \times 3$. If $\det ((a+1)\text{adj}((a-1)A))$ is $2^m 3^n$, $m, n \in \{0, 1, 2, \dots, 20\}$, then $m+n$ is equal to :
Important Questions from Algebra
Let A = {-3, -2, -1, 0, 1, 2, 3}. Let R be a relation on A defined by xRy if and only if $0\le x^2+2y\le 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $l + m$ is equal to
Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then $\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to
Let the domain of the function $f (x) = \log_2 \log_4 \log_6(3 + 4x -x^2)$ be $(a, b)$.
If $\int_{b-a}^{b+a}[x^2] dx=p-\sqrt{q}-\sqrt{r}$, $p,q,r\in N$, $gcd(p,q,r) = 1$, where $[.]$ is the greatest integer function, then $p + q + r$ is equal to
Let A be a matrix of order $3 \times 3$ and $|A| = 5$. If $|2\text{adj} (3A \text{adj} (2A))| = 2^\alpha \cdot 3^\beta \cdot 5^\gamma$, $\alpha, \beta, \gamma \in N$, then $\alpha + \beta + \gamma$ is equal to
Let $a_1, a_2, a_3,....$ be a G.P. of increasing positive numbers. If $a_3a_5 = 729$ and $a_2 + a_4 = \frac{111}{4}$, then $24 (a_1 + a_2 + a_3)$ is equal to
