For any complex number w- c - id- let w- - where i -1- Let - and - be real numbers such that for all complex numbers z - x - iy satisfying z-z - - - -4- the ordered pair x-y lies on the circle x2 - y2 - 5x - 3y - 4 -0- Then which of the following statements isare TRUE-

Given: nbsp; ( z+αz + β ) = π4 Mid point, 52= α β2 ⇒α+β=5 ⋯(1) and nbsp; Slope of line AC=3/2 5/2+β=m_1 (say) Slope of line BC=3/2 5/2+α=m_2 (say) Since ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Inequality of 2 Complex Numbers

For any compl...

Question

For any complex number $w = c + i d$ , let $arg (w) \in (- π, π]$ , where $i = \sqrt{- 1}$ . Let $α$ and $β$ be real numbers such that for all complex numbers $z = x + i y$ satisfying $arg (\frac{z + α}{z + β}) = \frac{π}{4}$ , the ordered pair $(x, y)$ lies on the circle $x^{2} + y^{2} + 5 x - 3 y + 4 = 0$ . Then which of the following statements is(are) TRUE?

α = - 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

α β = 4

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

α β = - 4

No worries! We‘ve got your back. Try BYJU‘S free classes today!

β = 4

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is D $β = 4$
Given:
$arg (\frac{z + α}{z + β}) = \frac{π}{4}$

Mid point, $\frac{- 5}{2} = \frac{- α - β}{2}$
$\Rightarrow α + β = 5 \dots (1)$
and
Slope of line $A C = \frac{3 / 2}{- 5 / 2 + β} = m_{1}$ (say)
Slope of line $B C = \frac{3 / 2}{- 5 / 2 + α} = m_{2}$ (say)
Since, $m_{1} \cdot m_{2} = - 1$
$∴ \frac{3 / 2}{- 5 / 2 + β} \cdot \frac{3 / 2}{- 5 / 2 + α} = - 1$
$\Rightarrow α β - \frac{5}{2} (α + β) + \frac{25}{4} = - \frac{9}{4}$
$\Rightarrow α β - \frac{25}{2} + \frac{25}{4} = - \frac{9}{4}$
$\Rightarrow α β = \frac{25}{4} - \frac{9}{4}$
$\Rightarrow α β = 4 \dots (2)$
From $(1)$ and $(2),$ we have
$α = 1, β = 4$ or $α = 4, β = 1$

Suggest Corrections

Similar questions

Q. Let

α

and

β

be the real numbers and

z

be a complex number. If

x^{2} + α z + β = 0

has two distinct non-real roots with

R e (x) = 1

, then it is necessary that

Q. Let

a, b, x

and

y

be real numbers such that

a - b = 1

and

y \neq 0

. If the complex number

z = x + i y

satisfies

I m (\frac{a z + b}{z + 1}) = y

, then which of the following is(are) possible value(s) of

x

Q. Let

z = x + i y

and

w = u + i v

be complex numbers on the unit circle such that

z^{2} + w^{2} = 1

. Then the number of ordered pairs

(z, w)

Q. Find the centre and radius of the circle formed by all the points represented by

z = x + i y

satisfying the relation

\frac{| z - α |}{| z - β |} = k (k \neq 1)

where

α

and

β

are constant complex numbers given by

α = α_{1} + {i α}_{2}

β = β_{1} + {i β}_{2}

Q. Let

a, b, x

and

y

be real numbers such that

a - b = 1

and

y \neq 0

. If the complex number

z = x + i y

satisfies

I m (\frac{a z + b}{z + 1}) = y

, then which of the following is(are) possible value(s) of

x

Join BYJU'S Learning Program

For any complex number w=c+id, let arg(w)∈(−π,π], where i=√−1. Let α and β be real numbers such that for all complex numbers z=x+iy satisfying arg(z+αz+β)=π4, the ordered pair (x,y) lies on the circle x2+y2+5x−3y+4=0. Then which of the following statements is(are) TRUE?