2.1(2): General Equation of a Circle | Brief Solutions

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(1) No real points; (2) the circle $(x-1)^2+(y-2)^2=5$; (3) the point $(1,2)$; (4) a circle only for $|a|>2$. Complete the square and check whether the squared radius is positive.

$(x-2)^2+(y-\tfrac34)^2=\tfrac{25}{16}$. The center lies on $x=2$ and is equidistant from all three points.

$t=-1$; the circle is $x^2+y^2-2x-1=0$. Substitution and Vieta give $\overrightarrow{OA}\cdot\overrightarrow{OB}=5x_1x_2=t$.

$x^2+y^2-2x-12=0$. Substitute the two points and use $-D-E=2$ to find $D=-2,E=0,F=-12$.

$x^2+y^2-5x-y+4=0$. Substitute the three points into the general equation.

$(x+a/2)^2+(y+1)^2=a^2/4-1$. A circle for $|a|>2$, the point $(-a/2,-1)$ for $|a|=2$, and no real points for $|a|<2$.

$a=-1$.

$x=1$;$x-\sqrt3y+2=0$.

$60^\circ$.

Tangent at the origin.

$x^2+y^2-ax-by=0$, obtained by substituting the three points.

For $\lambda=-1$, the line $y=x$; otherwise a circle with center $(\frac{\lambda}{1+\lambda},\frac1{1+\lambda})$ and radius $\frac{\sqrt{\lambda^2+1}}{|1+\lambda|}$.

$a\ne0$. Since $r^2=8(1/a-1/2)^2+2$, the minimum occurs at $a=2$: $(x-1)^2+(y+1)^2=2$.

For $k=1$, $x=0$; otherwise $(x-\frac{5(k^2+1)}{k^2-1})^2+y^2=\frac{100k^2}{(k^2-1)^2}$. Square the distance equation and complete the square.

(1) $-\frac17<t<1$; (2) $(x-\frac{24}{7})^2+(y+\frac{13}{49})^2=\frac{16}{7}$; (3) $t\in(-\frac17,0)\cup(\frac34,1)$. The center is $(t+3,4t^2-1)$ and $r^2=-7t^2+6t+1$; the outside condition is $t^2+1>r^2$.

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2.1(2): General Equation of a Circle | Brief Solutions

E1Decide whether the four quadratic equations represent circles.

E2Circle through $(1,0),(3,0),(2,2)$.

E3The circle $x^2+y^2-2x+t=0$ meets $2x+y=0$ with $\overrightarrow{OA}\cdot\overrightarrow{OB}=-1$.

E4Circle through $(4,2),(-1,3)$ whose four axis intercepts sum to $2$.

C1Circle through $(3,2),(1,1),(2,-1)$.

C2Classify $x^2+y^2+ax+2y+2=0$.

H1Find $a$ for which $a^2x^2+(a+2)y^2+2ax+a=0$ is a circle.

H2Tangents from $(1,\sqrt3)$ to the unit circle.

H3Angle between tangents from the origin to $x^2+y^2-12y+27=0$.

H4Relation between $Ax+By=0$ and the circle $x^2+y^2+Ax+By=0$.

H5Position of the origin relative to $x^2+y^2+2ax+2y+(a-1)^2=0$, $0<a<1$.

H6Symmetry of $x^2+y^2+2\sqrt2x-2\sqrt2y=0$.

H7Circle through the origin, $(a,0)$ and $(0,b)$.

H8Classify $x^2+y^2-2y+\lambda(x^2+y^2-2x)=0$.

H9What kind of condition is $A=C\ne0,B=0$ for a quadratic equation to represent a circle?

H10The circle $x^2+y^2+ax+by+c=0$ is symmetric about $x-2y=0$.

H11Circle condition and minimum-radius circle for $ax^2+ay^2-4(a-1)x+4y=0$.

H12Locus $PA=kPB$ for $A(-5,0),B(5,0)$ and $k>0$.

H13Parameterized circle: existence, maximum area, and when $P(3,4t^2)$ is outside.