1.7 Coordinate Geometry, Lines, Circles, and Conics

1.7 Coordinate Geometry, Lines, Circles, and Conics

Coordinate geometry converts pictures into equations. For points $P=(x_1,y_1)$ and $Q=(x_2,y_2)$, $d(P,Q)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2},\qquad M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$ A nonvertical line has slope $m=(y_2-y_1)/(x_2-x_1)$ and point-slope form $y-y_1=m(x-x_1)$. Parallel nonvertical lines have equal slopes; perpendicular slopes satisfy $m_1m_2=-1$. The vector form $ax+by=c$ has normal vector $(a,b)$, so the distance from $(x_0,y_0)$ to the line is $\frac{|ax_0+by_0-c|}{\sqrt{a^2+b^2}}.$

Circles and completing the square

A circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2+(y-k)^2=r^2$. To identify a circle from a general equation, group variables and complete squares. For $x^2+y^2-6x+4y-12=0,$ write $(x-3)^2-9+(y+2)^2-4=12$, so $(x-3)^2+(y+2)^2=25$. The center is $(3,-2)$ and radius is 5. A line is tangent to a circle exactly when its distance from the center equals the radius.

Standard conics

An ellipse centered at $(h,k)$ with horizontal major axis is $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1,\qquad a>b>0.$ Its foci are $(h\pm c,k)$ where $c^2=a^2-b^2$, and eccentricity is $e=c/a<1$. A circle is the special case $a=b$.

A hyperbola with horizontal transverse axis is $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1.$ Its vertices are $(h\pm a,k)$, foci are $(h\pm c,k)$ with $c^2=a^2+b^2$, and asymptotes are $y-k=\pm(b/a)(x-h)$. Its eccentricity exceeds 1.

A vertical-axis parabola has $(x-h)^2=4p(y-k)$. Its vertex is $(h,k)$, focus is $(h,k+p)$, directrix is $y=k-p$, and axis is $x=h$. The sign of $p$ gives the opening direction. A horizontal-axis parabola is $(y-k)^2=4p(x-h)$.

Classification of a general quadratic

For $Ax^2+Bxy+Cy^2+Dx+Ey+F=0,$ the discriminant $B^2-4AC$ classifies a nondegenerate real conic: negative indicates an ellipse, zero a parabola, and positive a hyperbola. The equation can still be degenerate or have no real points, so classification is only the first step. If $B\ne0$, a rotation of axes can remove the cross term; the rotation angle satisfies $\tan(2\theta)=B/(A-C)$ when $A\ne C$. Translation then completes squares without changing the conic type.

Intersections and tangency

Substitute a line into a conic to obtain a quadratic in one coordinate. Two distinct real roots mean two intersections, a repeated root means tangency, and no real roots mean no intersection. For the line $y=mx+b$ and unit circle $x^2+y^2=1$, substitution gives $(1+m^2)x^2+2mbx+(b^2-1)=0.$ Tangency requires discriminant zero. Equivalently, the distance from the origin to $mx-y+b=0$ must equal 1, giving $|b|/\sqrt{m^2+1}=1$.

Worked example

Identify $9x^2+4y^2-54x+8y+49=0$. Completing squares, $9[(x-3)^2-9]+4[(y+1)^2-1]+49=0,$ so $9(x-3)^2+4(y+1)^2=36$, or $\frac{(x-3)^2}{4}+\frac{(y+1)^2}{9}=1.$ This ellipse has center $(3,-1)$, vertical semimajor axis $a=3$, horizontal semiminor axis $b=2$, and $c=\sqrt5$. Its foci are $(3,-1\pm\sqrt5)$.

Common traps

The larger denominator determines an ellipse's major-axis direction. Hyperbola asymptotes pass through its center but are not part of the curve. In a parabola, the coefficient is $4p$, not $p$. A zero conic discriminant does not by itself rule out a degenerate pair of parallel lines, so simplify the whole equation when needed.

Vectors, loci, and degeneracy checks

Vector equations handle vertical lines and perpendicularity without separate slope cases. The line through $P$ with direction vector $\mathbf v$ is $\mathbf r(t)=P+t\mathbf v$. Two direction vectors are perpendicular exactly when their dot product is zero. For a line $ax+by=c$, the vector $(a,b)$ is normal, so a parallel line keeps the same left side and changes only the constant.

A locus description should be translated before expanding. Points equidistant from a fixed point and a fixed line form a parabola; points whose sum of distances to two foci is constant form an ellipse; a constant absolute difference of focal distances gives a hyperbola. These definitions explain the eccentricity classification: $e<1$ for an ellipse, $e=1$ for a parabola, and $e>1$ for a hyperbola.

The quadratic discriminant identifies a type but not whether the locus is nondegenerate. For instance, $x^2-y^2=0$ has positive $B^2-4AC$ yet factors as $(x-y)(x+y)=0$, a pair of intersecting lines rather than a hyperbola. Likewise, $x^2+y^2+1=0$ has no real points. After classifying, factor when possible or complete squares and inspect the resulting constants. Under a rotation, distances and angles are preserved, so geometric features may be easier to read in the rotated coordinates even though the cross term disappears.

Conic classification workflow

1. Read $A$, $B$, and $C$ from the complete quadratic and compute $B^2-4AC$.
2. Rotate axes if a cross term obstructs completing the square.
3. Translate and complete squares to expose the standard form.
4. Check signs and constants for an empty or degenerate locus.
5. Only then report the center or vertex, axis directions, foci, asymptotes, and eccentricity.