Lesson: Equation of a Plane in Space

In this lesson, we will learn how to find the equation of a plane using the x-, y-, and z-intercepts.

Worksheet: 15 Questions

Q1:

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 ) .

Q2:

Find the general equation of the plane that is perpendicular to the plane 6 𝑥 + 3 𝑦 + 4 𝑧 + 4 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 5 , 0 , 0 ) and ( 0 , 1 , 0 ) respectively.

Q3:

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.

Q4:

Given that the plane 2 𝑥 + 6 𝑦 + 2 𝑧 = 1 8 intersects the coordinate axes 𝑥 , 𝑦 , and 𝑧 at the points 𝐴 , 𝐵 , and 𝐶 , respectively, find the area of 𝐴 𝐵 𝐶 .

Q5:

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

Q6:

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

Q7:

Determine the general equation of the plane that intersects the negative 𝑥 -axis at a distance of 5 from the origin, intersects the negative 𝑧 -axis at a distance of 6 from the origin, and passes through the point 𝐶 ( 6 , 1 , 2 ) .

Q8:

Determine the general equation of the plane that intersects the positive 𝑥 -axis at a distance of 6 from the origin, intersects the negative 𝑧 -axis at a distance of 2 from the origin, and passes through the point 𝐶 ( 4 , 4 , 3 ) .

Q9:

Determine the general equation of the plane that intersects the negative 𝑥 -axis at a distance of 6 from the origin, intersects the negative 𝑧 -axis at a distance of 2 from the origin, and passes through the point 𝐶 ( 6 , 6 , 9 ) .

Q10:

Determine the general equation of the plane that intersects the negative 𝑥 -axis at a distance of 4 from the origin, intersects the positive 𝑧 -axis at a distance of 7 from the origin, and passes through the point 𝐶 ( 5 , 1 , 7 ) .

Q11:

Find the general equation of the plane that is perpendicular to the plane 8 𝑥 𝑦 + 6 𝑧 8 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 1 , 0 , 0 ) and ( 0 , 6 , 0 ) respectively.

Q12:

Find the general equation of the plane that is perpendicular to the plane 6 𝑥 5 𝑦 7 𝑧 + 4 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 1 , 0 , 0 ) and ( 0 , 1 , 0 ) respectively.

Q13:

Find the general equation of the plane that is perpendicular to the plane 3 𝑥 + 4 𝑦 2 𝑧 8 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 1 , 0 , 0 ) and ( 0 , 6 , 0 ) respectively.

Q14:

Find the general equation of the plane that is perpendicular to the plane 8 𝑥 5 𝑦 + 8 𝑧 + 5 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 4 , 0 , 0 ) and ( 0 , 3 , 0 ) respectively.

Q15:

Find the general equation of the plane that passes through the point ( 1 , 7 , 6 ) and cuts off equal intercepts on the three coordinate axes.

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