the equation of a plane in terms of the intercepts made by the plane on the coordinate axes. Let the equation of the plane be
Ax + By +Cz +D = 0 (D ≠ 0) ... (1)
Let the plane make intercepts a, b, c on x, y and z axes, respectively in following fig.
Hence, the plane meets x, y and z-axes at (a, 0, 0), (0, b, 0), (0, 0, c), respectively.
Therefore Aa + D = 0 or A = `(-D)/a`
Bb + D = 0 or B = `(-D)/b`
Cc +D = 0 or C = `(-D)/c`
Substituting these values in the equation (1) of the plane and simplifying, we get
`x/a + y/b +z/ c = 1` ...(1)
Shaalaa.com | 3 Dimensional Geometry part 21 (Plane in intercept form)
Find the equation of the plane which contains the line of intersection of the planes x \[+\] 2y \[+\] 3 \[z - \] 4 \[=\] 0 and 2 \[x + y - z\] \[+\] 5 \[=\] 0 and whose x-intercept is twice its z-intercept. Hence, write the equation of the plane passing through the point (2, 3, \[-\] 1) and parallel to the plane obtained above.
Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.
2x + 3y − z = 6
Find the equation of a plane which meets the axes at A, B and C, given that the centroid of the triangle ABC is the point (α, β, γ).
A plane meets the coordinate axes at A, B and C, respectively, such that the centroid of triangle ABC is (1, −2, 3). Find the equation of the plane.
Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then `1/a^2 + 1/b^2 + 1/c^2 = 1/p^2`