Circles & Lines

Study Sheet

Circles & Lines

Distances, angle bisectors, tangents, and conic meetings

Distance: Point to Line

Tip
The point-to-line distance formula

The distance from (x0,y0)(x_0, y_0) to the line Ax+By+C=0Ax + By + C = 0 is d=∣Ax0+By0+C∣A2+B2d = \dfrac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}; parallel lines Ax+By=C1Ax+By=C_1, Ax+By=C2Ax+By=C_2 sit ∣C1−C2∣A2+B2\dfrac{|C_1 - C_2|}{\sqrt{A^2+B^2}} apart.

In plain terms. Plug the point into the line's left side and divide by the length of (A,B)(A, B) — it measures how far the point sits along the line's perpendicular (normal) direction. Never solve for the foot of the perpendicular unless the problem asks for it.

Example. From (6,7)(6, 7) to 3x+4y−10=03x + 4y - 10 = 0: d=∣18+28−10∣5=365d = \dfrac{|18 + 28 - 10|}{5} = \dfrac{36}{5}.

Reminder — Distance and midpoint:  d=(x2−x1)2+(y2−y1)2,M=(x1+x22, y1+y22)d=\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)
Concept
Tangency is a distance statement

A line is tangent to a circle exactly when the center's distance to the line EQUALS the radius; it misses when the distance exceeds the radius, and cuts a chord when it is smaller.

In plain terms. One formula answers three different-looking questions: "is it tangent?", "what k makes it tangent?", and "how long is the chord?" (half-chord =r2−d2=\sqrt{r^2 - d^2}).

Example. y=x+ky = x + k is tangent to x2+y2=8x^2 + y^2 = 8 when ∣k∣2=22\dfrac{|k|}{\sqrt2} = 2\sqrt2, i.e. k=±4k = \pm 4.

The Bisector Between Two Lines

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Angle bisectors as an equidistance locus

The bisectors of the angles between A1x+B1y+C1=0A_1x + B_1y + C_1 = 0 and A2x+B2y+C2=0A_2x + B_2y + C_2 = 0 satisfy A1x+B1y+C1A12+B12=±A2x+B2y+C2A22+B22\dfrac{A_1x + B_1y + C_1}{\sqrt{A_1^2+B_1^2}} = \pm\dfrac{A_2x + B_2y + C_2}{\sqrt{A_2^2+B_2^2}} — two perpendicular lines, one per sign.

In plain terms. A bisector is just the set of points at EQUAL distance from both lines, so set the two distance formulas equal. The ±\pm gives both bisectors; pick the acute or obtuse one by testing a point.

Example. For y=34xy = \tfrac34x and the xx-axis: ∣3x−4y∣5=∣y∣\dfrac{|3x-4y|}{5} = |y| gives y=13xy = \tfrac13 x (acute) and y=−3xy = -3x (obtuse) — note the two bisectors are perpendicular.

Concept
Symmetry shortcut

If swapping x↔yx \leftrightarrow y (or another visible reflection) exchanges the two lines, then the mirror line of that reflection is automatically one of the bisectors.

In plain terms. Before computing, look for a symmetry that hands you the bisector for free.

Example. 3x+4y=123x + 4y = 12 and 4x+3y=124x + 3y = 12 swap under x↔yx \leftrightarrow y, so y=xy = x bisects them — no algebra needed.

Circles: Chords, Tangents, Power

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The three right-triangle facts

Chord at distance dd from the center: length 2r2−d22\sqrt{r^2 - d^2}. Tangent from an external point at distance dd: length d2−r2\sqrt{d^2 - r^2}. Common external tangent of circles (r1,r2)(r_1, r_2) with centers dd apart: d2−(r2−r1)2\sqrt{d^2 - (r_2 - r_1)^2} (internal: d2−(r1+r2)2\sqrt{d^2 - (r_1 + r_2)^2}).

In plain terms. Every one of these is the SAME move: drop the radius to the chord or tangency point, get a right triangle, apply Pythagoras. Learn the move, not four formulas.

Example. Tangent from (13,0)(13, 0) to x2+y2=25x^2 + y^2 = 25: 169−25=12\sqrt{169 - 25} = 12.

Reminder — The Pythagorean theorem:  a2+b2=c2a^2+b^2=c^2
Concept
Nearest point of a circle to a line

If the center is at distance d>rd > r from a line, the circle's nearest point to the line is at distance d−rd - r and the farthest at d+rd + r — both along the perpendicular from the center.

In plain terms. Circles turn min/max distance questions into "center distance, then adjust by the radius."

Example. From x2+y2=9x^2 + y^2 = 9 to 3x+4y=253x + 4y = 25: center distance 55, so the minimum is 5−3=25 - 3 = 2.

Circles Meet Parabolas

Tip
Intersecting a circle with a conic

Substitute the parabola into the circle and solve the resulting equation in u=x2u = x^2 (or yy): each POSITIVE root uu contributes two points (x=±ux = \pm\sqrt u), u=0u = 0 one point, and negative roots none.

In plain terms. The whole art is bookkeeping: a quadratic in uu can produce 0,1,2,3,0, 1, 2, 3, or 44 intersection points depending on the signs of its roots — count carefully instead of guessing from a sketch.

Example. y=x2−4y = x^2 - 4 into x2+y2=4x^2 + y^2 = 4: u2−7u+12=0u^2 - 7u + 12 = 0 gives u=3,4u = 3, 4, both positive — four intersection points.

Tip
Focus and directrix

The parabola x2=4pyx^2 = 4py has focus (0,p)(0, p) and directrix y=−py = -p; every point is equidistant from the two. In the form y=ax2y = ax^2, p=14ap = \dfrac{1}{4a}.

In plain terms. When a contest problem says "focus", convert to x2=4pyx^2 = 4py first — the geometry (equal distances) is what problems actually use.

Example. y=x28y = \tfrac{x^2}{8} is x2=8yx^2 = 8y: focus (0,2)(0, 2), directrix y=−2y = -2.