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Toolkit 95

Coordinate Geometry

Slope, Line Equation, and Distance

Two points on a line with slope, line equation, and distance formulas
mAB=y2−y1x2−x1m_{AB}=\frac{y_2-y_1}{x_2-x_1}
y−y1=m(x−x1)y-y_1=m(x-x_1)
AB=(x2−x1)2+(y2−y1)2AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Distance from a Point to a Line

Perpendicular distance from a point to a line
For P(x0,y0) and ℓ:ax+by+c=0,\text{For }P(x_0,y_0)\text{ and }\ell:ax+by+c=0,
d(P,ℓ)=∣ax0+by0+c∣a2+b2d(P,\ell)=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}

Distance from a Point to a Plane

Perpendicular distance from a point to a plane
For P(x0,y0,z0) and α:ax+by+cz+d=0,\text{For }P(x_0,y_0,z_0)\text{ and }\alpha:ax+by+cz+d=0,
d(P,α)=∣ax0+by0+cz0+d∣a2+b2+c2d(P,\alpha)=\frac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}}

Distance Between Two Parallel Lines

Distance between two parallel lines
ℓ1:ax+by+c1=0,ℓ2:ax+by+c2=0\ell_1:ax+by+c_1=0,\qquad \ell_2:ax+by+c_2=0
d(ℓ1,ℓ2)=∣c1−c2∣a2+b2d(\ell_1,\ell_2)=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}

Equation of a Circle

Circle with center and radius
A circle with center (x0,y0) and radius r:\text{A circle with center }(x_0,y_0)\text{ and radius }r:
(x−x0)2+(y−y0)2=r2(x-x_0)^2+(y-y_0)^2=r^2

Finding the Equation of a Tangent Line

Tangent line through an external point to a circle

Let the tangent line through the point P(x₀, y₀) have slope m.

ℓ: y−y0=m(x−x0)\ell:\ y-y_0=m(x-x_0)
(x−x1)2+(y−y1)2=r2(x-x_1)^2+(y-y_1)^2=r^2

Substitute the line equation into the circle equation. Since a tangent line intersects the circle at exactly one point, set the discriminant equal to zero and solve for m.

Δ=0\Delta=0
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