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

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.

  • A𝑥4𝑦7𝑧4=1
  • B𝑥3𝑦7𝑧4=1
  • C𝑥7𝑦4+𝑧3=1
  • D𝑥7+𝑦3𝑧4=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+2𝑧8=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=1𝑡+2𝑡, 𝑧=1+2𝑡+𝑡
  • D𝑥=1+2𝑡+𝑡, 𝑦=2𝑡𝑡, 𝑧=1+𝑡+2𝑡
  • E𝑥=1+𝑡+2𝑡, 𝑦=2𝑡𝑡, 𝑧=1+2𝑡+𝑡

Q4:

Which of the following is a parametric form of the equation of the plane that contains the line 𝑥+12=𝑦22=𝑧54 and the vector d=1,3,1?

  • A𝑥=1+2𝑡+5𝑡, 𝑦=2+2𝑡+3𝑡, 𝑧=5+4𝑡+𝑡
  • B𝑥=12𝑡+𝑡, 𝑦=2+2𝑡+3𝑡, 𝑧=5+4𝑡+𝑡
  • C𝑥=12𝑡+𝑡, 𝑦=2+2𝑡+3𝑡, 𝑧=5+4𝑡+𝑡
  • D𝑥=12𝑡+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 𝑥12=𝑦+11=𝑧13 and 𝑥4=𝑦22=𝑧+16?

  • A𝑥=12𝑡+𝑡, 𝑦=1𝑡+3𝑡, 𝑧=1+3𝑡+2𝑡
  • B𝑥=12𝑡𝑡, 𝑦=1𝑡+3𝑡, 𝑧=1+3𝑡2𝑡
  • C𝑥=𝑡+𝑡, 𝑦=4𝑡+𝑡, 𝑧=12𝑡
  • D𝑥=𝑡+𝑡, 𝑦=2𝑡3𝑡, 𝑧=12𝑡
  • E𝑥=12𝑡𝑡, 𝑦=1𝑡+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=12𝑡𝑡, 𝑦=5+𝑡+2𝑡, 𝑧=1+3𝑡+2𝑡
  • B𝑥=22𝑡𝑡, 𝑦=3+4𝑡, 𝑧=4+2𝑡+3𝑡
  • C𝑥=2+2𝑡+𝑡, 𝑦=3𝑡+𝑡, 𝑧=4+2𝑡+𝑡
  • D𝑥=32𝑡𝑡, 𝑦=4+𝑡𝑡, 𝑧=32𝑡+𝑡
  • E𝑥=1𝑡+2𝑡, 𝑦=1+2𝑡+𝑡, 𝑧=1+3𝑡+2𝑡

Q7:

Write, in intercept form, the equation of the plane 16𝑥+2𝑦+8𝑧16=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=34𝑡, 𝑧=1+3𝑡.

  • A𝑥+4𝑦+7𝑧+16=0
  • B𝑥+12𝑦28𝑧=0
  • C3𝑥+3𝑦7𝑧+4=0
  • D12𝑥+4𝑦+7𝑧43=0
  • E𝑥12𝑦+28𝑧16=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.

  • A2711
  • B219
  • C27112
  • D3192
  • E3152

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