Worksheet: Equation of a Plane: Intercept and Parametric Forms

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

Q1:

Find the general equation of the plane 𝑥=4+7𝑡+4𝑡, 𝑦=−3−4𝑡, 𝑧=1+3𝑡.

  • A 𝑥 + 4 𝑦 + 7 𝑧 + 1 6 = 0
  • B 𝑥 + 1 2 𝑦 − 2 8 𝑧 = 0
  • C 3 𝑥 + 3 𝑦 − 7 𝑧 + 4 = 0
  • D 1 2 𝑥 + 4 𝑦 + 7 𝑧 − 4 3 = 0
  • E 𝑥 − 1 2 𝑦 + 2 8 𝑧 − 1 6 = 0

Q2:

Write, in intercept form, the equation of the plane 16𝑥+2𝑦+8𝑧−16=0.

  • A 𝑥 1 + 𝑦 8 + 𝑧 2 = 1 6
  • B 𝑥 1 6 + 𝑦 2 + 𝑧 8 = 1
  • C 𝑥 1 + 𝑦 8 + 𝑧 2 = 1
  • D 𝑥 1 6 + 𝑦 1 6 + 𝑧 1 6 = 1
  • E 𝑥 1 6 + 𝑦 2 + 𝑧 8 = 1 6

Q3:

What is the length of the segment of the 𝑦-axis cut off by the plane 5𝑥−4𝑦−3𝑧+32=0?

  • A 1 8 of a length unit
  • B8 length units
  • C32 length units
  • D4 length units

Q4:

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).

  • A 4 𝑥 + 𝑦 + 2 𝑧 + 8 = 0
  • B 𝑥 + 4 𝑦 + 2 𝑧 + 7 = 0
  • C 4 𝑥 + 𝑦 + 2 𝑧 − 8 = 0
  • D 𝑥 + 4 𝑦 + 2 𝑧 = 0

Q5:

Given that the plane 2𝑥+6𝑦+2𝑧=18 intersects the coordinate axes 𝑥, 𝑦, and 𝑧 at the points 𝐴, 𝐵, and 𝐶, respectively, find the area of △𝐴𝐵𝐶.

  • A 2 7 √ 1 1
  • B 2 √ 1 9
  • C 2 7 √ 1 1 2
  • D 3 √ 1 9 2
  • E 3 √ 1 5 2

Q6:

Determine the general equation of the plane that intersects the negative 𝑥-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 𝐶(9,−4,−4).

  • A 2 𝑥 + 3 𝑧 − 6 = 0
  • B 1 2 𝑥 + 4 1 𝑦 − 8 𝑧 + 2 4 = 0
  • C 1 1 𝑥 − 4 𝑦 − 4 𝑧 + 1 2 = 0
  • D 1 2 𝑥 − 4 1 𝑦 − 8 𝑧 − 2 4 = 0
  • E 9 𝑥 − 4 𝑦 − 7 𝑧 + 1 8 = 0

Q7:

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

  • A 𝑥 + 𝑦 + 𝑧 + 1 0 = 0
  • B 𝑥 + 𝑦 + 𝑧 − 6 4 8 = 0
  • C 8 𝑥 − 9 𝑦 − 9 𝑧 = 0
  • D 𝑥 + 𝑦 + 𝑧 − 1 0 = 0
  • E 8 𝑥 + 𝑦 + 𝑧 = 0

Q8:

Find the equation of the plane cutting the coordinate axes at 𝐴, 𝐵, and 𝐶, given that the intersection point of the medians of △𝐴𝐵𝐶 is (𝑙,𝑚,𝑛).

  • A 𝑙 𝑥 + 𝑚 𝑦 + 𝑛 𝑧 = 1
  • B 𝑥 + 𝑦 + 𝑧 = 𝑙 + 𝑚 + 𝑛
  • C 𝑙 𝑥 + 𝑚 𝑦 + 𝑛 𝑧 = 3
  • D 𝑥 𝑙 + 𝑦 𝑚 + 𝑧 𝑛 = 3
  • E 𝑥 𝑙 + 𝑦 𝑚 + 𝑧 𝑛 = 1

Q9:

Find the equation of the plane whose 𝑥-, 𝑦-, and 𝑧-intercepts are −7, 3, and −4, respectively.

  • A − 𝑥 4 − 𝑦 7 − 𝑧 4 = 1
  • B 𝑥 3 − 𝑦 7 − 𝑧 4 = 1
  • C − 𝑥 7 − 𝑦 4 + 𝑧 3 = 1
  • D − 𝑥 7 + 𝑦 3 − 𝑧 4 = 1

Q10:

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

  • A 4 3
  • B 2 3
  • C 3 2
  • D 4 5
  • E 5 4

Q11:

Choose the possible parametric equation of the plane 𝑥+2𝑦−3𝑧=3.

  • A 𝑥 = 3 𝑡 , 𝑦 = 3 𝑡 , 𝑧 = 1 + 𝑡 − 2 𝑡
  • B 𝑥 = − 3 𝑡 , 𝑦 = 3 𝑡 , 𝑧 = 𝑡 + 2 𝑡 − 1
  • C 𝑥 = 3 𝑡 , 𝑦 = 3 𝑡 , 𝑧 = 𝑡 + 2 𝑡 − 1
  • D 𝑥 = 3 𝑡 , 𝑦 = 3 𝑡 , 𝑧 = 1 − 𝑡 + 2 𝑡
  • E 𝑥 = 3 𝑡 , 𝑦 = − 3 𝑡 , 𝑧 = 𝑡 + 2 𝑡 − 1

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