2.3(1): Standard Equation of a Hyperbola | Brief Solutions

Use the summaries to locate questions in the original paper. Select a question or its button to reveal the answer. Calculations and proofs include the essential steps.

Labels: E = worked example, C = class practice, H = homework; P1-P4 refer to original PDF pages. Other labels follow the original question or section numbering.

$a=2,c=3,b^2=5$. Either $x^2/4-y^2/5=1$ with foci $(\pm3,0)$, or $y^2/4-x^2/5=1$ with foci $(0,\pm3)$.

$x^2/4-y^2/5=1,\ x\ge2$. The positive difference selects the right branch.

$y=0,\ x\ge3$, by equality in the triangle inequality.

Set $A=(-1000,0),B=(1000,0)$. Then $PB-PA=1360$, giving $x^2/462400-y^2/537600=1,\ x\le-680$, in meters.

$x^2/(1/2)-y^2/(7/2)=1$. Internal and external tangency give absolute focal-distance difference $\sqrt2$, so $a=\sqrt2/2,c=2$.

$y=0,x\ge5$, by equality in the triangle inequality.

$x^2/16-y^2/9=1,x\ge4$. Use $a=4,c=5$ and the right branch.

$x^2/9-y^2/16=1,x\ge3$. Use $a=3,c=5$ and the right branch.

The question is incomplete; omitted.

$9$. From $r_1^2+r_2^2=100$ and $(r_1-r_2)^2=64$, the product is $18$ and the area is $9$.

$x^2/9-y^2/7=1$

$x^2/64-y^2/36=1,\quad x\ge8$

The question is incomplete; omitted.

$m<-2$

$x^2/16-y^2/9=1$;$y^2/16-x^2/9=1$

$41/4$

$9$. Since $PM\le PF_1+2$ and $PN\ge PF_2-1$, the difference is at most $2a+3=9$, attained by collinear choices.

$x^2/4-y^2/5=1,x\ge2$. External tangency gives $AB-AP=4$, selecting the right branch.

Aim N$30.0^\circ$E (E$60.0^\circ$N) from $A$. With $A=(3,0),B=(-3,0),C=(-5,2\sqrt3)$, solve $PB=PC$ and $PB-PA=4$ to get $P=(8,5\sqrt3)$.

A, C; $x=\pm2\sqrt3/3$

$(0,4)$ for the upper branch and $(0,-4)$ for the lower. Tangent lengths satisfy $u+v=10,|u-v|=8$, so $\{u,v\}=\{9,1\}$. Exclude vertices, where the triangle degenerates.

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2.3(1): Standard Equation of a Hyperbola | Brief Solutions

E1Focal distance $6$ and absolute focal-distance difference $4$: find standard equations and foci.

E2.1Distance to $(-3,0)$ minus distance to $(3,0)$ is $4$. Find the locus.

E2.2Same foci, with distance difference $6$.

E3Stations are $2000$ m apart; $A$ hears an explosion $4$ s before $B$ at $340$ m/s. Find the locus.

E4A moving circle passes through $(2,0)$ and is tangent to $(x+2)^2+y^2=2$. Find its center locus.

C1.1Foci $(-5,0),(5,0)$ and $|MF_1|-|MF_2|=10$.

C1.2Same foci, distance difference $8$.

C1.3Same foci, distance difference $6$.

C2Find $m$ from focal distance $10$; a denominator in the equation is missing.

C3A right focal angle on $x^2/16-y^2/9=1$: find the focal-triangle area.

H1Absolute distance difference to $(4,0),(-4,0)$ is $6$: find the locus.

H2Distance to $(-10,0)$ minus distance to $(10,0)$ is $16$.

H3Find a standard hyperbola through two points; the first x-coordinate is missing.

H4Find the parameter for confocal conics; a hyperbola denominator is missing.

H5$x^2/(m+2)-y^2/(m+1)=1$ is a hyperbola with foci on the y-axis.

H6Center at the origin, foci on a coordinate axis, transverse axis $8$, focal distance $10$: find equations.

H7For $x^2/16-y^2/9=1$, find the distance to the left focus from an intersection on the vertical through the right focus.

H8$x^2/4+y^2/a^2=1$ and $x^2/a^2-y^2/2=1$ are confocal. Find $a$.

H9$P$ is on the right branch of $x^2/9-y^2/16=1$. Points $M,N$ lie on circles centered at $(-5,0),(5,0)$ with radii $2,1$. Maximize $PM-PN$.

H10A moving circle passes through $P(3,0)$ and is externally tangent to $(x+3)^2+y^2=16$. Find its center locus.

H11Locate a signal source: $A$ is $6$ km east of $B$, $C$ is $4$ km N$30^\circ$W of $B$, and $B,C$ receive the signal $4$ s after $A$ at $1$ km/s.

H12Solve $|\sqrt{x^2+4x+5}-\sqrt{x^2-4x+5}|=2$.

H13$P$ lies on $y^2/16-x^2/9=1$. Find where the focal-triangle incircle touches the focal segment.