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MATH / For 0<k<6, the graphs of ((x−k)²)/(9)+y²=1 and (x²)/(9)+y²=1 intersect at A and C, and have x-intercepts at B and D respectively. Compute the value of k for which ABCD is a square.…

Problem

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

problem

For 0<k<6,0 < k < 6, the graphs of (x−k)29+y2=1\frac{(x - k)^2}{9} + y^2 = 1 and x29+y2=1\frac{x^2}{9} + y^2 = 1 intersect at AA and C,C, and have xx-intercepts at BB and DD respectively. Compute the value of kk for which ABCDABCD is a square. [asy] unitsize(1 cm); path ellone = xscale(3)*Circle((0,0),1); path elltwo = shift((24/5,0))*xscale(3)*Circle((0,0),1); pair A, B, C, D; A = intersectionpoints(ellone,elltwo)[0]; C = intersectionpoints(ellone,elltwo)[1]; B = (-3 + 24/5,0); D = (3,0); draw(ellone); draw(elltwo); draw((-3.5,0)--(9,0)); draw((0,-1.5)--(0,1.5)); draw(A--B--C--D--cycle); label("AA", A, N, fontsize(10)); label("BB", B, NW, fontsize(10)); label("CC", C, S, fontsize(10)); label("DD", D, NE, fontsize(10)); [/asy]
Plain-text mathematical notation (without MathML)
For 0<k<6, the graphs of ((x−k)²)/(9)+y²=1 and (x²)/(9)+y²=1 intersect at A and C, and have x-intercepts at B and D respectively.  Compute the value of k for which ABCD is a square.

[asy]
unitsize(1 cm);

path ellone = xscale(3)*Circle((0,0),1);
path elltwo = shift((24/5,0))*xscale(3)*Circle((0,0),1);
pair A, B, C, D;

A = intersectionpoints(ellone,elltwo)[0];
C = intersectionpoints(ellone,elltwo)[1];
B = (-3 + 24/5,0);
D = (3,0);

draw(ellone);
draw(elltwo);

draw((-3.5,0)--(9,0));
draw((0,-1.5)--(0,1.5));
draw(A--B--C--D--cycle);

label("A", A, N, fontsize(10));
label("B", B, NW, fontsize(10));
label("C", C, S, fontsize(10));
label("D", D, NE, fontsize(10));
[/asy]
Original LaTeX notation
For $0 < k < 6,$ the graphs of $\frac{(x - k)^2}{9} + y^2 = 1$ and $\frac{x^2}{9} + y^2 = 1$ intersect at $A$ and $C,$ and have $x$-intercepts at $B$ and $D$ respectively.  Compute the value of $k$ for which $ABCD$ is a square.

[asy]
unitsize(1 cm);

path ellone = xscale(3)*Circle((0,0),1);
path elltwo = shift((24/5,0))*xscale(3)*Circle((0,0),1);
pair A, B, C, D;

A = intersectionpoints(ellone,elltwo)[0];
C = intersectionpoints(ellone,elltwo)[1];
B = (-3 + 24/5,0);
D = (3,0);

draw(ellone);
draw(elltwo);

draw((-3.5,0)--(9,0));
draw((0,-1.5)--(0,1.5));
draw(A--B--C--D--cycle);

label("$A$", A, N, fontsize(10));
label("$B$", B, NW, fontsize(10));
label("$C$", C, S, fontsize(10));
label("$D$", D, NE, fontsize(10));
[/asy]

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
bd3c1dd8-c5c4-5f76-a066-a1321590c824
task_key
intermediate~5falgebra--test--bd3c1dd8-c5c4-5f76-a066-a1321590c824
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import