Lesson Worksheet: Equation of a Plane: Intercept and Parametric Forms Mathematics

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

  • A−𝑥4−𝑦7−𝑧4=1
  • B𝑥3−𝑦7−𝑧4=1
  • C−𝑥7−𝑦4+𝑧3=1
  • D−𝑥7+𝑦3−𝑧4=1

Q2:

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).

  • A4𝑥+𝑦+2𝑧+8=0
  • B𝑥+4𝑦+2𝑧+7=0
  • C4𝑥+𝑦+2𝑧−8=0
  • D𝑥+4𝑦+2𝑧=0

Q3:

Find, in parametric form, the equation of the plane that passes through the point 𝐴(1,2,1) and contains the two vectors d=⟨1,−1,2⟩ and d=⟨2,−1,1⟩.

  • A𝑥=1+𝑡+2𝑡, 𝑦=2+2𝑡+𝑡, 𝑧=1+2𝑡+𝑡
  • B𝑥=1+2𝑡+𝑡, 𝑦=2+2𝑡+𝑡, 𝑧=1+𝑡+2𝑡
  • C𝑥=1+𝑡+2𝑡, 𝑦=1−𝑡+2𝑡, 𝑧=1+2𝑡+𝑡
  • D𝑥=1+2𝑡+𝑡, 𝑦=2−𝑡−𝑡, 𝑧=1+𝑡+2𝑡
  • E𝑥=1+𝑡+2𝑡, 𝑦=2−𝑡−𝑡, 𝑧=1+2𝑡+𝑡

Q4:

Which of the following is a parametric form of the equation of the plane that contains the line 𝑥+1−2=𝑦−22=𝑧−54 and the vector d=⟨1,3,1⟩?

  • A𝑥=−1+2𝑡+5𝑡, 𝑦=2+2𝑡+3𝑡, 𝑧=−5+4𝑡+𝑡
  • B𝑥=−1−2𝑡+𝑡, 𝑦=2+2𝑡+3𝑡, 𝑧=−5+4𝑡+𝑡
  • C𝑥=−1−2𝑡+𝑡, 𝑦=−2+2𝑡+3𝑡, 𝑧=5+4𝑡+𝑡
  • D𝑥=−1−2𝑡+𝑡, 𝑦=2+2𝑡+3𝑡, 𝑧=5+4𝑡+𝑡
  • E𝑥=−1+2𝑡+5𝑡, 𝑦=−2+2𝑡+3𝑡, 𝑧=5+4𝑡+𝑡

Q5:

Which of the following is the parametric form of the equation of the plane that contains the two lines 𝑥−1−2=𝑦+1−1=𝑧−13 and 𝑥−4=𝑦−2−2=𝑧+16?

  • A𝑥=1−2𝑡+𝑡, 𝑦=−1−𝑡+3𝑡, 𝑧=1+3𝑡+2𝑡
  • B𝑥=1−2𝑡−𝑡, 𝑦=−1−𝑡+3𝑡, 𝑧=1+3𝑡−2𝑡
  • C𝑥=−𝑡+𝑡, 𝑦=−4𝑡+𝑡, 𝑧=−1−2𝑡
  • D𝑥=−𝑡+𝑡, 𝑦=2−𝑡−3𝑡, 𝑧=−1−2𝑡
  • E𝑥=1−2𝑡−𝑡, 𝑦=−1−𝑡+3𝑡, 𝑧=−1+3𝑡

Q6:

Find the parametric form of the equation of the plane that passes through the points 𝐴(1,5,1), 𝐵(3,4,3), and 𝐶(2,3,4).

  • A𝑥=1−2𝑡−𝑡, 𝑦=5+𝑡+2𝑡, 𝑧=1+3𝑡+2𝑡
  • B𝑥=2−2𝑡−𝑡, 𝑦=3+4𝑡, 𝑧=4+2𝑡+3𝑡
  • C𝑥=2+2𝑡+𝑡, 𝑦=3−𝑡+𝑡, 𝑧=4+2𝑡+𝑡
  • D𝑥=3−2𝑡−𝑡, 𝑦=4+𝑡−𝑡, 𝑧=3−2𝑡+𝑡
  • E𝑥=−1−𝑡+2𝑡, 𝑦=−1+2𝑡+𝑡, 𝑧=1+3𝑡+2𝑡

Q7:

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

  • A𝑥1+𝑦8+𝑧2=16
  • B𝑥16+𝑦2+𝑧8=1
  • C𝑥1+𝑦8+𝑧2=1
  • D𝑥16+𝑦16+𝑧16=1
  • E𝑥16+𝑦2+𝑧8=16

Q8:

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

  • A𝑥+4𝑦+7𝑧+16=0
  • B𝑥+12𝑦−28𝑧=0
  • C3𝑥+3𝑦−7𝑧+4=0
  • D12𝑥+4𝑦+7𝑧−43=0
  • E𝑥−12𝑦+28𝑧−16=0

Q9:

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

  • A43
  • B23
  • C32
  • D45
  • E54

Q10:

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.

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

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