Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

How many units from (0,0) is the point on the circle (x+12)^2 + (y-5)^2 = 16 that is closest to (0,0)?

A)10

B)9

C)8.5

D)7.5

E)6.5

How is the answer is B? It would be great if someone explained. Thanks:)

3 Answers

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  • 1 month ago
    Favourite answer

    the circle is centered at –12,+5

    so the line from 0,0 that passes thru the center will also yield the closest point.

    line is y = kx

    5 = –12k

    k = –5/12

    equation is y = –(5/12)x

    where does this intersect the circle?

    (x+12)² + (y-5)² = 16

    (x+12)² + (–(5/12)x-5)² = 16

    (x+12)² + ((5/12)x+5)² = 16

    x² + 24x + 144 + (25/144)x² + (25/6)x + 25 – 16 = 0

    x²(144/144+25/144) + x(144/6+25/6) + 153 = 0

    x²(169/144) + x(169/6) + 153 = 0

    using a quadratic solver

    x = –8.3076, –15.6923

    using the one closest to the origin, x =  –8.3076

    y = –(5/12)x = 3.4615

    and distance = √(8.3076² + 3.4615²) = 8.9999 (B)

    method 2 (much easier):

    distance from 0,0 to center of circle is √(12² + 5²) = 13

    distance from center of circle to circumference = √16 = 4

    difference is 9 units

    Attachment image
  • Center of the circle is at (-12 , 5)

    Circle has a radius that is sqrt(16), or 4

    Find the distance from the center of the circle to the origin

    sqrt((-12 - 0)^2 + (5 - 0)^2) =>

    sqrt(144 + 25) =>

    sqrt(169) =>

    13

    13 - 4 = 9

    There you go.  Similarly, the point on the circle furthest from the origin would be 13 + 4, or 17

  • 1 month ago

    Have you tried graphing the situation? Check the graph in the link below.

    Remember that a circle with center (h,k) and radius r can be written as:

    (x - h)² + (y - k)² = r²

    Looking at that and comparing it to your equation, you can quickly figure out that the center of the circle is (-12, 5) and the radius is 4.

    Next, using your knowledge of Pythagorean Triples (e.g. 3,4,5 or 5,12,13) you should be able to figure out that the center is 13 units from the origin. If that wasn't readily apparent, use the distance formula between two points.

    d = √[(-12 - 0)² + (5 - 0)²]

    d = √(144 + 25)

    d = √169

    d = 13

    In summary the distance from the center to the origin is 13 units.

    The radius of the circle is 4 units.

    If you subtract you get the distance from the edge of the circle (closest point) to the origin.

    13 - 4 = 9 units

    Answer:

    B) 9

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