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Properties of Modulus

Let x , y , z...

Question

Let $x, y, z \in C$ satisfy $| x | = 1, | y - 6 - 8 i | = 3$ and $| z + 1 - 7 i | = 5$ respectively, then the minimum value of $| x - z | + | y - z |$ is equal to

1

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2

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5

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6

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Solution

The correct option is D $6$
$| z_{1} + z_{2} | \leq | z_{1} | + | z_{2} | \forall z_{1}, z_{2} \in C$

$| x - z | + | y - z | \geq | (x - z) + (z - y) | = | x - y |$

Equality holds when $x, y, z$ are collinear points. (The points are collinear on the line $3 y = 4 x$ in the Argand plane)

$| x - y |_{m i n} = C_{1} C_{2} - (r_{1} + r_{2}) = 10 - (1 + 3) = 6$

Suggest Corrections

Similar questions

x, y, z \in C

| x | = 1, | y - 6 - 8 i | = 3

| z + 1 - 7 i | = 5

| x - z | + | y - z |

\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}

\frac{x + 1}{- 1} = \frac{y - 2}{α} = \frac{z - 5}{5}

α

Δ_{1} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} y^{5} z^{6} (z^{3} - y^{3}) & x^{4} z^{6} (x^{3} - z^{3}) & x^{4} y^{5} (y^{3} - x^{3}) y^{2} z^{3} (y^{6} - z^{6}) & x z^{3} (z^{6} - x^{6}) & x y^{2} (x^{6} - y^{6}) y^{2} z^{3} (z^{3} - y^{3}) & x z^{3} (x^{3} - z^{3}) & x y^{2} (y^{3} - x^{3}) \end{matrix} ∣ ∣ ∣ ∣ ∣

Δ_{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} x & y^{2} & z^{3} x^{4} & y^{5} & z^{6} x^{7} & y^{8} & z^{9} \end{matrix} ∣ ∣ ∣ ∣ ∣

Δ_{1}

x, y, z

\neq 0, \neq 1

∣ ∣ ∣ ∣ \begin{matrix} log x & log y & log z log 2 x & log 2 y & log 2 z log 3 x & log 3 y & log 3 z \end{matrix} ∣ ∣ ∣ ∣

{xN}_{2} + {yH}_{2} \to {zNH}_{3}

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