2.1(3): Relative Positions of a Line and a Circle | Brief Solutions

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Two for $|b|<3\sqrt5$, one for equality, none for $|b|>3\sqrt5$. Compare $d=|b|/\sqrt5$ with $r=3$.

$4\sqrt5$, using center $(3,1)$, $r=5$, $d=\sqrt5$ and $L=2\sqrt{r^2-d^2}$.

$x=-3$ or $3x+4y+15=0$. The center-to-line distance is $3$; include the vertical case.

$(x-1)^2+(y+3)^2=10$, with $x^2+y^2<16$. Use $\overrightarrow{OC}\cdot\overrightarrow{MC}=0$ and a nonzero chord.

(1) Every line passes through the interior point $(3,1)$, hence has two intersections. (2) The shortest chord is on $2x-y-5=0$ ($m=-3/4$), perpendicular to the center-to-fixed-point segment; its length is $4\sqrt5$.

$\sqrt2$, since $d=1/\sqrt2$ and $L=2\sqrt{1-d^2}$.

$b=\pm3\sqrt2$, since $b^2/2=25-4^2=9$.

$(x-3)^2+(y-4)^2=25$, with $x^2+y^2<16$, by the perpendicular from the center to the chord.

(1) $-2<a<2$; (2) $a<-2$ or $a>2$; (3) $a=\pm2$.

$0<k<\frac43$.

$x+y-2=0$ or $7x+17y+26=0$.

$x^2+(y-\tfrac12)^2=\tfrac14$, excluding $(0,0),(0,1)$. The perpendicular relation gives the circle; $k=0$ is tangent and the vertical secant is absent from the family.

$m=\pm\frac{\sqrt6}{3}$, from $m^2+\left(\frac{2\sqrt3}{3}\right)^2=2$.

$3x+4y+20=0$ or $3x+4y-10=0$. The center is $(1,-2)$, radius $5$, and required line distance $3$.

$(x-3)^2+(y-1)^2=9$ or $(x+3)^2+(y+1)^2=9$. Set the center to $(3t,t)$; $9t^2-2t^2=7$ gives $t=\pm1$.

$y=x+1$ or $y=x-4$. With $y=x+b$, the condition $\overrightarrow{OA}\cdot\overrightarrow{OB}=0$ gives $b^2+3b-4=0$; both lines are secants.

Every line passes through the interior point $(3,1)$, giving two intersections. The shortest chord is on $2x-y-5=0$ and has length $4\sqrt5$.

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2.1(3): Relative Positions of a Line and a Circle | Brief Solutions

E1Number of intersections of $2x-y+b=0$ and $x^2+y^2=9$.

E2.1Chord cut by $x+2y=0$ in $x^2+y^2-6x-2y-15=0$.

E2.2Lines through $(-3,-\tfrac32)$ cutting a length-8 chord in $x^2+y^2=25$.

E3Chord-midpoint locus for secants from $M(2,-6)$ to $x^2+y^2=16$.

E4Parameterized lines $(2m+1)x+(m+1)y-7m-4=0$ and circle $(x-1)^2+(y-2)^2=25$.

C1Chord cut by $x-y-5=0$ in $(x-2)^2+(y+2)^2=1$.

C2$x-y+b=0$ cuts a length-8 chord in $x^2+y^2=25$.

C3Chord-midpoint locus for secants from $(6,8)$ to $x^2+y^2=16$.

H1Secant, disjoint and tangent conditions for $x+y-a=0$ and $x^2+y^2=2$.

H2$y=kx+2$ has two distinct intersections with $(x-2)^2+(y-3)^2=1$.

H3Lines through $(6,-4)$ cutting a chord of length $6\sqrt2$ in $x^2+y^2=20$.

H4Chord-midpoint locus for $kx-y+1=0$ in the unit circle.

H5$x\sin\alpha+y\cos\alpha+m=0$ cuts a chord of length $4\sqrt3/3$ in $x^2+y^2=2$.

H6A chord on $ax+y-2=0$ and the center of $(x-1)^2+(y-a)^2=4$ form an equilateral triangle.

H7$y=x+m$ cuts $x^2+y^2=16$ with $|MN|\ge\sqrt3|\overrightarrow{OM}+\overrightarrow{ON}|$.

H8Number of points on $x^2+y^2+2x+4y-3=0$ at distance $\sqrt2$ from $x+y+1=0$.

H9Lines perpendicular to $4x-3y+18=0$ cutting a length-8 chord in $x^2+y^2-2x+4y-20=0$.

H10Circle tangent to the $y$ axis, center on $x-3y=0$, chord length $2\sqrt7$ on $y=x$.

H11Slope-1 secants of $x^2+y^2-2x+4y-4=0$ whose chord-diameter circle passes through the origin.

H12Same as Example 4: two intersections and the shortest chord.