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Geometry Level 1

Which of these equations describes a circle of radius 5, and centered at (3, 4)?

( x + 3 ) 2 + ( y + 4 ) 2 = 25 (x+3)^2 + (y+4)^2 = 25 ( x − 3 ) 2 + ( y − 4 ) 2 = 5 (x-3)^2 + (y-4)^2 = 5 ( x − 3 ) 2 + ( y − 4 ) 2 = 25 (x-3)^2 + (y-4)^2 = 25 ( x + 3 ) 2 + ( y + 4 ) 2 = 5 (x+3)^2 + (y+4)^2 = 5

Matilda Belanno by Matilda Belanno , 40, USA

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2 solutions

center of circle is known=(3 , 4). as it passes through origin ,the distance between center and origin is equal to the radius of the circle . so radius =sqrt[ (3^2)+(4^2)] =5 equation of circle with center (a ,b) and radius r is given by (x-a)^2 +(y-b)^2 =r^2. hance 3rd option is correct answer.

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Andy Wong
Oct 8, 2015

All answers are in the form ( x − h ) 2 + ( y − k ) 2 = r 2 { (x-h) }^{ 2 }\quad +\quad { (y-k) }^{ 2 }\quad =\quad { r }^{ 2 } , with (h, k) as the center and r as the radius. If you plug in the given values into this equation, you will find that the equation is ( x − 3 ) 2 + ( y − 4 ) 2 = 5 2 { (x-3) }^{ 2 }\quad +\quad { (y-4) }^{ 2 }\quad =\quad { 5 }^{ 2 }

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If you are not regularly using or happen to forget the formula, then you can use test points to determine which of the formulas above make sense. If there is a diagram, such as this problem, you can just check which points on the drawn circle satisfy the above equations.

Examples: setting x=3 (ie. no x-component), y=-1 or y=9 to be part of the circle and only the second equation yields this.

Scott Ripperda - 5 years, 8 months ago

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