Question

Circle O is represented by the equation (x + 7)2 + (y + 7)2 = 16. What is the length of the radius of circle O?
A. 3
B. 4
C. 7
D. 9
E. 16

Answer 1

GIVEN :–

$\: { \huge{. \: \: }}{ \rm{Equation \: \: of \: \: circle \: - {(x + 7)}^{2} + {(y + 7) }^{2} = 16 }}$

TO FIND :–

$\: { \huge{. \: \: }}{ \rm{radius = ? }}$

SOLUTION :–

METHOD - (1)

• If radius of circle is r and center of circle is (a , b)

then standard equation of circle is –

$\\ { \huge{. \: \: }}{ \green{ \boxed{ \bold{ {(x - a)}^{2} + {(y - b) }^{2} = {r}^{2} }}}}$

• So that –

$\\ \implies{ \rm{ {(x - ( - 7))}^{2} + {(y - ( - 7)) }^{2} = {(4)}^{2} }}$

• Now compare –

$\\ \: { \huge{ \: \: . \: \: }}{ \rm{ a = - 7}}$

$\\ \: { \huge{ \: \: . \: \: }}{ \rm{ b= - 7}}$

$\\ \: { \huge{ \: \: . \: \: }}{ \rm{ r= 4}}$

Hence, Center is (-7 , -7) and radius(r) = 4

METHOD - (2) :–

$\\ { \huge{ . \: }}{ \rm{ Genral \: \: equation \: \: of \: \: circle \: - }}$

$\\ { \huge{ \: \: \star \: \: }} \large{ \green{ \boxed{ \bold{ {x }^{2} + {y}^{2} + 2gx + 2fy + c = 0 }}}}$

$\\ { \huge{.}}{ \rm{ \: \: Here \: \: - }}$

$\\ \: \: \: \: \: \: \: \: .{ \pink { \rm{ \: \: \: \: center = ( - g, - f)}}}$

$\\ \: \: \: \: \: \: \: \: .{ \pink{ \rm{ \: \: \: \: radius = \sqrt{ {g}^{2} + {f}^{2} - c } }}}$

• So that –

$\\ \implies{ \rm{ {(x + 7)}^{2} + {(y + 7)}^{2} = 16}}$

• Using identity –

$\\ \bigstar \: \large{ \red{ \boxed{ \rm{ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab }}}}$

$\\ \implies{ \rm{ {x}^{2} + {7}^{2} + 2(x)(7)+ {y}^{2} + {7}^{2} + 2(y)(7) = 16}}$

$\\ \implies{ \rm{ {x}^{2} + {y}^{2} + 14x + 14y + 49 + 49 = 16}}$

$\\ \implies{ \rm{ {x}^{2} + {y}^{2} + 14x + 14y + 98 - 16= 0}}$

$\\ \implies{ \rm{ {x}^{2} + {y}^{2} + 14x + 14y + 82= 0}}$

• Now compare –

$\\ \: \: \: \: \: \: \: \: . { \rm{ \: \: \: \: g = 7}}$

$\\ \: \: \: \: \: \: \: \: . { \rm{ \: \: \: \: f = 7}}$

$\\ \: \: \: \: \: \: \: \: . { \rm{ \: \: \: \: c = 82}}$

• So that –

$\\ \: \: \: .{ { \rm{ \: \: \: \: center = ( - 7, - 7)}}}$

$\\ \: \: \: .{ { \rm{ \: \: \: \: radius = \sqrt{ {(7)}^{2} + {(7)}^{2} - 82 } }}}$

$\\ \implies{ { \rm{ \: radius = \sqrt{ 49+ 49 - 82 } }}}$

$\\ \implies{ { \rm{ \: radius = \sqrt{ 98 - 82 } }}}$

$\\ \implies{ { \rm{ \: radius = \sqrt{ 16 } }}}$

$\\ \implies{ { \rm{ \: radius = \pm \: 4 }}}$

• But radius will be positive , so that –

$\\ \implies{ \red{ \boxed { \rm{ \: radius = 4 }}}}$

Hence , Option (b) is correct.

Answer 2

Given ,

The eq of circle is :

(x + 7)² + (y + 7)² = 16

It can be written as ,

$\sf \Rightarrow {(x - (-7))}^{2} + { (y - (-7))}^{2} = {(4)}^{2} \: --- \: (i)$

We know that , the standard eq of circle is given by

$\sf \star \: \: {(x - h)}^{2} + {(y - k)}^{2} = {(r)}^{2}$

Where ,

• (h,k) = centre of circle
• (x,y) = any point on the perimeter of circle
• (r) = radius of circle

On comparing eq (i) with standard eq of the circle , we get

h = -7

k = -7

r = 4

$\therefore \sf \bold{ \underline{The \: radius \: of \: circle \: is \: 4 \: units }}$