Question

Let z be a complex number then the area bounded by the curve |z-5i|^(2)+|z-12|^(2)=169 in square units is​

Answer 1

SOLUTION

GIVEN

Let z be a complex number and the equation of the curve is

$\sf{ { |z - 5i| }^{2} + { |z - 12| }^{2} = 169 }$

TO DETERMINE

The area bounded by the curve

EVALUATION

Here the given equation of the curve is

$\sf{ { |z - 5i| }^{2} + { |z - 12| }^{2} = 169 }$

Let z = x + iy

Then above curve can be rewritten as below

$\sf{ { |(x + iy) - 5i| }^{2} + { |(x + iy) - 12| }^{2} = 169 }$

$\sf{ \implies \: { |x + i(y - 5)| }^{2} + { |(x - 12) + iy| }^{2} = 169 }$

$\sf{ \implies \: { \bigg( \sqrt{ {x}^{2} + {(y - 5)}^{2} } \bigg)}^{2} + { \bigg( \sqrt{ {(x - 12)}^{2} + {y}^{2} } \bigg)}^{2} = 169 }$

$\sf{ \implies \: { {x}^{2} + {(y - 5)}^{2} } + {(x - 12)}^{2} + {y}^{2} = 169 }$

$\sf{ \implies \: {x}^{2} + {y }^{2} - 10y + 25+ {x}^{2} - 24x + 144 + {y}^{2} = 169 }$

$\sf{ \implies \: 2 {x}^{2} +2 {y }^{2} - 24x- 10y = 0 }$

$\sf{ \implies \: {x}^{2} + {y }^{2} - 12x- 5y = 0 }$

$\displaystyle \sf{ \implies \: {x}^{2} - 2.x.6 + {(6)}^{2} + {y }^{2} - 2.y. \frac{5}{2} + { \bigg( \frac{5}{2} \bigg)}^{2} = {6}^{2} +{ \bigg( \frac{5}{2} \bigg)}^{2} }$

$\displaystyle \sf{ \implies \: {(x - 6)}^{2} + { \bigg( y - \frac{5}{2} \bigg)}^{2} = 36 + \frac{25}{4} }$

$\displaystyle \sf{ \implies \: {(x - 6)}^{2} + { \bigg( y - \frac{5}{2} \bigg)}^{2} = \frac{169}{4} }$

$\displaystyle \sf{ \implies \: {(x - 6)}^{2} + { \bigg( y - \frac{5}{2} \bigg)}^{2} = { \bigg( \frac{13}{2} \bigg)}^{2} }$

The above equation represents a circle with origin

$\displaystyle \sf{ at \: \bigg(6, \frac{5}{2} \bigg) \: \: and \: \: radius \: = \frac{13}{2} \: \: unit}$

Hence area of the curve

= Area of the circle

$\displaystyle \sf{ = \pi \: \times { \bigg( \frac{13}{2} \bigg)}^{2} \: \: sq \: unit}$

$\displaystyle \sf{ = \frac{169\pi}{4} \: \: sq \: unit}$

$\displaystyle \sf{ =42.25\pi \: \: sq \: unit}$

$\displaystyle \sf{ = 132.79 \: \: sq \: unit}$

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Additional Information :

Complex Number

A complex number z = a + ib is defined as an ordered pair of Real numbers ( a, b) that satisfies the following conditions :

(i) Condition for equality :

(a, b) = (c, d) if and only if a = c, b = d

(ii) Definition of addition :

(a, b) + (c, d) = (a+c, b+ d)

(iii) Definition of multiplication :

(a, b). (c, d) = (ac-bd , ad+bc )

Of the ordered pair (a, b) the first component a is called Real part of z and the second component b is called Imaginary part of z

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