Worksheet: Equation of a Plane: Intercept and Parametric Forms

In this worksheet, we will practice finding the equation of a plane in different forms, such as intercept and parametric forms.


Find the general equation of the plane π‘₯=4+7𝑑+4π‘‘οŠ§οŠ¨, 𝑦=βˆ’3βˆ’4π‘‘οŠ¨, 𝑧=1+3π‘‘οŠ§.

  • Aπ‘₯+4𝑦+7𝑧+16=0
  • Bπ‘₯+12π‘¦βˆ’28𝑧=0
  • C3π‘₯+3π‘¦βˆ’7𝑧+4=0
  • D12π‘₯+4𝑦+7π‘§βˆ’43=0
  • Eπ‘₯βˆ’12𝑦+28π‘§βˆ’16=0


Write, in intercept form, the equation of the plane 16π‘₯+2𝑦+8π‘§βˆ’16=0.

  • Aπ‘₯1+𝑦8+𝑧2=16
  • Bπ‘₯16+𝑦2+𝑧8=1
  • Cπ‘₯1+𝑦8+𝑧2=1
  • Dπ‘₯16+𝑦16+𝑧16=1
  • Eπ‘₯16+𝑦2+𝑧8=16


What is the length of the segment of the 𝑦-axis cut off by the plane 5π‘₯βˆ’4π‘¦βˆ’3𝑧+32=0?

  • A18 of a length unit
  • B8 length units
  • C32 length units
  • D4 length units


Find the general form of the equation of the plane which intersects the coordinate axes at the points (2,0,0), (0,8,0), and (0,0,4).

  • A4π‘₯+𝑦+2𝑧+8=0
  • Bπ‘₯+4𝑦+2𝑧+7=0
  • C4π‘₯+𝑦+2π‘§βˆ’8=0
  • Dπ‘₯+4𝑦+2𝑧=0


Given that the plane 2π‘₯+6𝑦+2𝑧=18 intersects the coordinate axes π‘₯, 𝑦, and 𝑧 at the points 𝐴, 𝐡, and 𝐢, respectively, find the area of △𝐴𝐡𝐢.

  • A27√11
  • B2√19
  • C27√112
  • D3√192
  • E3√152


Determine the general equation of the plane that intersects the negativeΒ π‘₯-axis at a distance of 2 from the origin, intersects the positive 𝑧-axis at a distance of 3 from the origin, and passes through the point 𝐢(9,βˆ’4,βˆ’4).

  • A2π‘₯+3π‘§βˆ’6=0
  • B12π‘₯+41π‘¦βˆ’8𝑧+24=0
  • C11π‘₯βˆ’4π‘¦βˆ’4𝑧+12=0
  • D12π‘₯βˆ’41π‘¦βˆ’8π‘§βˆ’24=0
  • E9π‘₯βˆ’4π‘¦βˆ’7𝑧+18=0


Find the general equation of the plane that passes through the point (8,βˆ’9,βˆ’9) and cuts off equal intercepts on the three coordinate axes.

  • Aπ‘₯+𝑦+𝑧+10=0
  • Bπ‘₯+𝑦+π‘§βˆ’648=0
  • C8π‘₯βˆ’9π‘¦βˆ’9𝑧=0
  • Dπ‘₯+𝑦+π‘§βˆ’10=0
  • E8π‘₯+𝑦+𝑧=0


Find the equation of the plane cutting the coordinate axes at 𝐴, 𝐡, and 𝐢, given that the intersection point of the medians of △𝐴𝐡𝐢 is (𝑙,π‘š,𝑛).

  • A𝑙π‘₯+π‘šπ‘¦+𝑛𝑧=1
  • Bπ‘₯+𝑦+𝑧=𝑙+π‘š+𝑛
  • C𝑙π‘₯+π‘šπ‘¦+𝑛𝑧=3
  • Dπ‘₯𝑙+π‘¦π‘š+𝑧𝑛=3
  • Eπ‘₯𝑙+π‘¦π‘š+𝑧𝑛=1


Find the equation of the plane whose π‘₯-, 𝑦-, and 𝑧-intercepts are βˆ’7, 3, and βˆ’4, respectively.

  • Aβˆ’π‘₯4βˆ’π‘¦7βˆ’π‘§4=1
  • Bπ‘₯3βˆ’π‘¦7βˆ’π‘§4=1
  • Cβˆ’π‘₯7βˆ’π‘¦4+𝑧3=1
  • Dβˆ’π‘₯7+𝑦3βˆ’π‘§4=1


What is the length of the segment of the π‘₯-axis cut off by the plane 6π‘₯+3𝑦+5𝑧=4?

  • A43
  • B23
  • C32
  • D45
  • E54


Choose the possible parametric equation of the plane π‘₯+2π‘¦βˆ’3𝑧=3.

  • Aπ‘₯=3π‘‘οŠ§, 𝑦=3π‘‘οŠ¨, 𝑧=1+π‘‘βˆ’2π‘‘οŠ§οŠ¨
  • Bπ‘₯=βˆ’3π‘‘οŠ§, 𝑦=3π‘‘οŠ¨, 𝑧=𝑑+2π‘‘βˆ’1
  • Cπ‘₯=3π‘‘οŠ§, 𝑦=3π‘‘οŠ¨, 𝑧=𝑑+2π‘‘βˆ’1
  • Dπ‘₯=3π‘‘οŠ§, 𝑦=3π‘‘οŠ¨, 𝑧=1βˆ’π‘‘+2π‘‘οŠ§οŠ¨
  • Eπ‘₯=3π‘‘οŠ§, 𝑦=βˆ’3π‘‘οŠ¨, 𝑧=𝑑+2π‘‘βˆ’1

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