Prove that the equation of circle in the z plane can be written in the from - zz- - z-z- -c-0 Deduce the equation of the line-

The correct option is A β̅ z+βz̅ +c=0 If z=x+iy then z̅ =x iy ∴ x= z+z̅ 2 ,y= z z̅ 2i ,zz̅ =| z | ^ 2 = x ^ 2+ y ^ 2 The equation of a c ...

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Byju's Answer

Standard XII

Mathematics

Equation of a Plane Passing through a Point and Perpendicular to a Given Vector

Prove that th...

Question

Prove that the equation of circle in the $z$ plane can be written in the from $α z ¯ z + ¯ β z + β ¯ z + c = 0$ Deduce the equation of the line.

¯ β z + β ¯ z + c = 0

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¯ β z - β ¯ z + c = 0

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¯ β z + β ¯ z - c = 0

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None of these

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Solution

The correct option is A $¯ β z + β ¯ z + c = 0$
If $z = x + i y$ then $¯ z = x - i y$
$∴ x = \frac{z + ¯ z}{2}, y = \frac{z - ¯ z}{2 i}, z ¯ z = {| z |}^{2} = x^{2} + y^{2}$
The equation of a circle is
$α (x^{2} + y^{2}) + 2 g x + 2 f y + c = 0$
$α z ¯ z + g (z + ¯ z) + f \frac{(z - ¯ z)}{i} + c = 0$
or $α z ¯ z + g (z + ¯ z) - i f (z - ¯ z) + c = 0$
or $α z ¯ z + (g - i f) z + (g + i f) ¯ z + c = 0$
Now if $g + i f = β$ then $g - i f = ¯ β$
$∴ α z ¯ z + ¯ β z + β ¯ z + c = 0$
It is of the same form as given.
Deduction: Circle $(1)$ will reduce to a straight line if $α = 0$ in $(1)$ . Hence putting $α = 0$ in $(2)$ the equation of the line can be put in the form
$¯ β z + β ¯ z + c = 0$

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Similar questions

(i) If

A (z_{1}), B (z_{2})

are two complex numbers, then distance between

z_{1}

z_{2}

A B = | z_{1} - z_{2} |

(ii) Line segment joining the points

A (z_{1}), B (z_{2})

is divided by the point

P (z)

in the ratio

m : n

, then

z = \frac{m z_{2} + n z_{1}}{m + n}

where

m, n

are real numbers.

(iii) Points

z_{1}, z_{2}, z_{3}

are collinear if

∣ ∣ ∣ ∣ \begin{matrix} z_{1} & _{1} & 1 z_{2} & _{2} & 1 z_{3} & _{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0

and the general equation of line joining

A (z_{1}), B (z_{2})

in non-parametric given by

∣ ∣ ∣ ∣ \begin{matrix} z & ¯ z & 1 z_{1} & _{1} & 1 z_{2} & _{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0

and general equation of straight line is

a ¯ z + ¯ a z + b = 0

where

a

is a complex number &

b

is real, the real slope of the line is

\frac{a + ¯ a}{i (¯ a - a)}

or equal to

\frac{- R e (a)}{I m g (a)}

and the complex slope of the line is

- \frac{a}{¯ a}

\frac{coefficient of ¯ z}{coefficient of z}

.The lines

¯ a z + a ¯ z + λ = 0

and

z ¯ a - ¯ z a + i λ = 0

are respectively parallel and

⊥

to the line

a ¯ z + ¯ a z + b = 0

(iv) The length of

⊥

from a point

z_{1}

to the line

¯ a z + a ¯ z + b = 0

is given by

\frac{| a_{1} + ¯ a z_{1} + b |}{2 | a |}

(v) If the point

A (z_{1}), B (z_{2})

are to be considered as foci and length of major axis is

2 a

then equation of ellipse is given by

| z - z_{1} | + | z - z_{2} | = 2 a ⟹ 2 a > | z_{1} - z_{2} |

If major axis is to be considered as transversal axis with length

2 a

such that

2 a < | z_{1} - z_{2} |

The equation of hyperbola is given by

∥ z - z_{1} ∣ - ∣ z - z_{2} ∥ = 2 a

(vi) If

| z - α | = r

(α

is complex) is a circle at

α = (p, q)

& radius

= r

Let

α, β

be two fixed non zero complex numbers & '

z

' a variable complex number such that the lines

α ¯ z + ¯ α z + 1 = 0

and

β ¯ z + ¯ β z + 1 = 0

are mutually perpendicular lines, then value of

α ¯ β + β ¯ α

equals

Q. Prove that the equation of a line through two given points in the Argand plane can be put in the form

z ¯ β + ¯ z β + c = 0

where

β

is non-zero complex and

c

is real.

Q. The equation of the plane through the intersection of the planes ax + by + cz + d = 0 and lx + my + nz + p = 0 and parallel to the line y=0, z=0
(a) (bl − am) y + (cl − an) z + dl − ap = 0
(b) (am − bl) x + (mc − bn) z + md − bp = 0
(c) (na − cl) x + (bn − cm) y + nd − cp = 0
(d) None of these

Q. Equation y-axis is considered as
(a) x = 0, y = 0
(b) y = 0, z = 0
(c) z = 0, x = 0
(d) none of these

Q. If the roots of the equation:

a x^{2} + b x + c = 0, a > 0

be each greater than unity, then:

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Prove that the equation of circle in the z plane can be written in the from αz¯z+¯βz+β¯z+c=0 Deduce the equation of the line.

Prove that the equation of circle in the $z$ plane can be written in the from $α z ¯ z + ¯ β z + β ¯ z + c = 0$ Deduce the equation of the line.