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Find the number of common tangents to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$.
Circle 1: centre $C_1=(2,3)$, $r_1=\sqrt{4+9+12}=5$.
Circle 2: centre $C_2=(-3,-9)$, $r_2=\sqrt{9+81-26}=8$.
Distance $|C_1C_2|=\sqrt{(2+3)^2+(3+9)^2}=\sqrt{25+144}=\sqrt{169}=13$.
Since $|C_1C_2|=13=r_1+r_2=5+8$, the circles are externally tangent. Externally tangent circles have exactly 3 common tangents.
Negative space (also called void or counter-form) in architecture and design refers to the unoccupied space around, between, and within built forms.
It is as important as positive (built) space in defining the quality of architecture:
- The courtyard of a building is negative space that organises surrounding rooms.
- The gap between two buildings creates a negative space that functions as a street or plaza.
- Frank Lloyd Wright's concept: 'The reality of a building is the space within.'
Understanding negative space is fundamental to spatial composition in architecture and urban design.
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