benchmarks.wiki

benchmarks.wiki / Public workspace

MATH / Two circles, one centered at (−3,2) and the other centered at (0,−1), are internally tangent as shown. [asy] import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Two circles, one centered at (−3,2)(-3,2) and the other centered at (0,−1)(0,-1), are internally tangent as shown. [asy] import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-9.34,xmax=9.27,ymin=-9.36,ymax=7.89; Label laxis; laxis.p=fontsize(10); xaxis(xmin,xmax,Ticks(laxis,Step=2.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis(ymin,ymax,Ticks(laxis,Step=2.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw(circle((0,-1),7.07)); draw(circle((-3,2),2.83)); dot((0,-1),ds); label("(0,−1)(0, -1)",(0.23,-1.87),SE*lsf); dot((-3,2),ds); label("(−3,2)(-3, 2)",(-2.82,2.29),N*lsf); dot((1,6),ds); label("(1,6)(1, 6)",(1.2,6.3),NE*lsf); dot((-5,4),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] If the equation of the smaller circle can be written as x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0, find D+E+FD + E + F.
Plain-text mathematical notation (without MathML)
Two circles, one centered at (−3,2) and the other centered at (0,−1), are internally tangent as shown. [asy]
import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-9.34,xmax=9.27,ymin=-9.36,ymax=7.89;

Label laxis; laxis.p=fontsize(10);

xaxis(xmin,xmax,Ticks(laxis,Step=2.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis(ymin,ymax,Ticks(laxis,Step=2.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw(circle((0,-1),7.07)); draw(circle((-3,2),2.83));

dot((0,-1),ds); label("(0,−1)",(0.23,-1.87),SE*lsf); dot((-3,2),ds); label("(−3,2)",(-2.82,2.29),N*lsf); dot((1,6),ds); label("(1,6)",(1.2,6.3),NE*lsf); dot((-5,4),ds);

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);

[/asy] If the equation of the smaller circle can be written as x²+y²+Dx+Ey+F=0, find D+E+F.
Original LaTeX notation
Two circles, one centered at $(-3,2)$ and the other centered at $(0,-1)$, are internally tangent as shown. [asy]
import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-9.34,xmax=9.27,ymin=-9.36,ymax=7.89;

Label laxis; laxis.p=fontsize(10);

xaxis(xmin,xmax,Ticks(laxis,Step=2.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis(ymin,ymax,Ticks(laxis,Step=2.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw(circle((0,-1),7.07)); draw(circle((-3,2),2.83));

dot((0,-1),ds); label("$(0, -1)$",(0.23,-1.87),SE*lsf); dot((-3,2),ds); label("$(-3, 2)$",(-2.82,2.29),N*lsf); dot((1,6),ds); label("$(1, 6)$",(1.2,6.3),NE*lsf); dot((-5,4),ds);

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);

[/asy] If the equation of the smaller circle can be written as $x^2 + y^2 + Dx + Ey + F = 0$, find $D + E + F$.

level

Level 5

type

Algebra

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
726a77d4-4599-5066-aba1-cd38278fdcbb
task_key
algebra--test--726a77d4-4599-5066-aba1-cd38278fdcbb
task_revision_id
3
config / split
algebra / test

Official source

initial import