Name: Reverse Triangle Inequality for Complex Numbers
Uploaded: 2019-07-19
Duration: 5 min 1 s
Description: In this interesting video, let us study and prove the triangle inequality associated with the difference of two complex numbers. Happy learning!

Question

If

{|z|}^{2} + \bar{A} z^{2} + A {\bar{z}}^{2} + B \bar{z} + \bar{B} z + c = 0

represents a pair of intersecting lines with angle of intersection

θ

, then find the value of

|A|

.

Accepted Answer

Let $z = x + i y, A = α + i β$ and $B = γ + i δ$

Given equation is,

${|z|}^{2} + \bar{A} z^{2} + A {\bar{z}}^{2} + B \bar{z} + \bar{B} z + c = 0$

Keeping the Cartesian form for all complex number $z, A & B$ in given equation, we get,

$x^{2} + y^{2} + 2 R e (A {\bar{z}}^{2}) + 2 R e (B \bar{z}) + c = 0$ $(∵ z + \bar{z} = 2 R e (z))$

or, $x^{2} + y^{2} + 2 R e ((α + i β) (x^{2} - y^{2} - 2 i x y)) + 2 R e ((γ + i δ) (x^{2} - y^{2} + 2 i x y)) + c = 0$

or, $x^{2} + y^{2} + 2 (α (x^{2} - y^{2}) + 2 β x y) + 2 (γ x + δ y) + c = 0$

or, $(1 + 2 α) x^{2} + 4 β x y + (1 - 2 α) y^{2} + 2 γ x + 2 δ y + c = 0 \dots (i)$

which is the form of general equation of second degree $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ representing a pair of lines.

Angle between a general pair of lines $(θ)$ is,

$\tan θ = \frac{2 \sqrt{h^{2} - a b}}{|a + b|}$

Comparing with equation $(i)$ , we get

$a = (1 + 2 α), h = 2 β$ and $b = (1 - 2 α)$

Hence, $\tan θ = \frac{2 \sqrt{4 β^{2} - (1 + 2 α) (1 - 2 α)}}{|(1 + 2 α) + (1 - 2 α)|}$

or, $\tan θ = \frac{2 \sqrt{4 β^{2} - (1 - 4 α^{2})}}{|2|}$

or, $\tan^{2} θ = 4 β^{2} + 4 α^{2} - 1 = 4 {|A|}^{2} - 1$

Hence, $4 {|A|}^{2} = 1 + \tan^{2} θ = s e c^{2} θ$

$\Rightarrow |A| = \frac{|\sec θ|}{2}$

If ${|z|}^{2} + \bar{A} z^{2} + A {\bar{z}}^{2} + B \bar{z} + \bar{B} z + c = 0$ represents a pair of intersecting lines with angle of intersection $θ$ , then find the value of $|A|$ .

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

If z2+A¯z2+Az¯2+Bz¯+B¯z+c=0 represents a pair of intersecting lines with angle of intersection θ, then find the value of A.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

If ${|z|}^{2} + \bar{A} z^{2} + A {\bar{z}}^{2} + B \bar{z} + \bar{B} z + c = 0$ represents a pair of intersecting lines with angle of intersection $θ$ , then find the value of $|A|$ .