Two circles have the following equations: ${(x + 6)}^{2} + {(y - 5)}^{2} = 64$ $(x +6 )^2+(y -5 )^2= 64$ and ${(x - 9)}^{2} + {(y + 4)}^{2} = 81$ $(x -9 )^2+(y +4 )^2= 81$ . Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

Question

Two circles have the following equations: ${(x + 6)}^{2} + {(y - 5)}^{2} = 64$ $(x +6 )^2+(y -5 )^2= 64$ and ${(x - 9)}^{2} + {(y + 4)}^{2} = 81$ $(x -9 )^2+(y +4 )^2= 81$ . Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

1 Answer

Impact of this question

1623 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-04-06T15:46:35

The circles overlap but the small circle isn't contained in the big one. The biggest possible distance between 2 points in the 2 circles is $\sqrt{226} + 17 ≅ 32.033$ $sqrt226+17~=32.033$

Explanation:

${(x + 6)}^{2} + {(y - 5)}^{2} = 64$ $(x+6)^2+(y-5)^2=64$ => $r_{1} = 8$ $r_1=8$ , $C_{1} (- 6, 5)$ $C_1 (-6,5)$
${(x - 9)}^{2} + {(y - 4)}^{2} = 81$ $(x-9)^2+(y-4)^2=81$ => $r_{2} = 9$ $r_2=9$ , $C_{2} (9, 4)$ $C_2 (9,4)$

$r_{1} + r_{2} = 17$ $r_1+r_2=17$
$r_{2} - r_{1} = 1$ $r_2-r_1=1$

$d_{C_{1} C_{2}} = \sqrt{{(9 + 6)}^{2} + {(4 - 5)}^{2}} = \sqrt{225 + 1} = \sqrt{226} ≅ 15.033$ $d_(C_1C_2)=sqrt((9+6)^2+(4-5)^2)=sqrt(225+1)=sqrt(226)~=15.033$

Since $r_{2} - r_{1} < d_{C_{1} C_{2}} < r_{1} + r_{2}$ $r_2-r_1 < d_(C_1C_2) < r_1+r_2$
the circles overlap but circle 1 isn't contained in circle 2.

The greatest possible distance between 2 points in the 2 circles is
$= d_{C_{1} C_{2}} + r_{1} + r_{2} = \sqrt{226} + 8 + 9 = \sqrt{226} + 17 ≅ 32.033$ $=d_(C_1C_2)+r_1+r_2=sqrt226+8+9=sqrt226+17~=32.033$

Science

Math

Humanities

... and beyond

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

Two circles have the following equations: (x+6)2+(y−5)2=64(x +6 )^2+(y -5 )^2= 64 and (x−9)2+(y+4)2=81(x -9 )^2+(y +4 )^2= 81 . Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

1 Answer

Explanation:

Related questions

Impact of this question