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8. PLANES
Synopsis :
1. Let 'P' and 'Q' be two points on a surface, if every point on
the line 'PQ' lies on the surface thenthat surface is called a
plane.
2. If a,b,c, are the d.rs of a normal to the plane passing
through the point ( )1 1 1, , x y z then the equation
of that plane is ax +by+cz = 1 1 1ax by cz+ +
CARTESIAN EQUATION OF A PLANE PASSING THROUGH THREE POINTS:
3. Equation of the plane passing through the points ( )1 1 1, ,
x y z ( )2 2 2, , x y z and ( )3 3 3, , x y z
is
1 1 1
2 1 2 1 2 1
3 1 3 1 3 1
0
x x y y z z
x x y y z z
x x y y z z
=
GENERAL EQUATION OF A PLANE:
4. The equation ax + by + cz + d = 0 represents a plane. Here a,
b,c are the d.rs of a normal to the
plane
5. The equation of a plane passing through the line of
intersection of the planes u = 0 and v = 0 is
0u v+ = where ' ' is a variable
6. Equations of the planes passing through the point (a,b,c) and
parallel to the yz plane, zx plane, xy plane are x = a, y =b, z = c
respectively
7. Equations ax+by+r=0, by+cz+p=0, cz+ax+q=0 represents the
planes perpendicular to XY, YZ,ZX planes respectively.
8. The foot of the perpendicular of the point ( )1 1 1, ,P x y z
on the plane 0ax by cz d + + + = is Q
( ), ,h k l then( )1 1 11 1 1
2 2 2
ax by cz d h x k y l z
a b c a b c
+ + + = = =
+ +
9. If Q (h, k, l) is the image of the point ( )1 1 1, ,p x y z
w.r.t plane 0ax by cz d + + + = then
( )1 1 11 1 12 2 2
2 ax by cz d h x k y l z
a b c a b c
+ + + = = =
+ +
10. Equations of two planes bisecting the angles between the
planes 1 1 1 1 0a x b y c z d + + + = and
2 2 2 2 0a x b y c z d + + + = are1 1 1 1 2 2 2 2
2 2 2 2 2 2
1 1 1 2 2 2
a x b y c z d a x b y c z d
a b c a b c
+ + + + + +=
+ + + +
NOTE : Make the constants 1 2,d d positive
Condition Acute Obtuse
i) 1 2 1 2 1 2 0a a b b c c+ + > +
ii) 1 2 1 2 1 2 0a a b b c c+ + < +
Note : i) Bisectors are perpendicular to each other
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8/9/2019 IB-08Planes(36-40)
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ii) Positive sign bisector is the bisector containing the
origin.
11. Equation of plane parallel to the plane 0ax by cz d + + +
=
is given by 0ax by cz k + + + =
12. The equation of the plane, mid way between the parallel
planes 1 0 Ax By Cz D+ + + = and
2 0 Ax By Cz D+ + + = is1 2 0
2
D D Ax By Cz
+ + + + =
13. The equation of the plane parallel to
I) x axis is of the form by + cz =d
II) y aixs is of the form ax + cz = d
III) z axis is of the form ax + by = d
14. The equation of the plane passing through ( )1 1 1, ,x y z
and parallel to
I) yz plane is 1x x=
II) zx plane is 1y y=
III) xy plane is1
z z=
15. The ratio in which the plane ax + by + cz + d = 0 divides
the line segment joining ( )1 1 1, , x y z and
( )2 2 2, ,x y z is( )
( )1 1 1
2 2 2
ax by cz d
ax by cz d
+ + +
+ + +
i)If1 1 1
ax by cz d + + + and 2 2 2ax by cz d + + + have the same sign
then ( )1 1 1, , x y z and ( )2 2 2, , x y z
lie on the same side of the plane.
ii)If 1 1 1ax by cz d + + + and 2 2 2ax by cz d + + + have
opposite signs, then ( )1 1 1, , x y z and
( )2 2 2, ,x y z lie on opposite sides of the plane.
Normal Form of a Plane:
16. Let OM be the perpendicular from O(0,0,0) to a plane. If OM
= P and l,m,n are the d.cs of OMthen the equation of that plane in
the Normal form is lx + my + nz = P
17. The perpendicular distance from ( )1 1 1, , x y z to the
plane ax + by + cz + d = 0 is1 1 1
2 2 2
ax by cz d
a b c
+ + +
+ +
18. The perpendicular distance of the palne ax+by+cz+d=0 from
the origin is2 2 2
d
a b c+ +.
19. The distance between the parallel planes ax + by + cz + dl=
0 and ax + by + cz + d
2= 0 is
1 2
2 2 2
d d
a b c
+ +
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Intercept Form of a Plane:
20. If a plane intersects the x, y, z axes at A,B,C respectively
and O = (0,0,0) then OA, OB,OC are
called the x intercept, y intercept, z intercept of the plane
respectively
21. The x,y,z intercepts of the plane ax + by + cz + d = 0 are ,
,d d d
a b c
respectively
22. If a, b, c are the intercepts of a plane then the equation
of the plane in the intercept form is
1 x y z
a b c+ + = .
23. The equation of the plane whose intercepts are 'K" times the
intercepts made by the plane
0 Ax By Cz D+ + + =
( )0D
is 0 Ax By Cz KD+ + + =
24. Area of the triangle formed by the plane 1 x y z
a b c+ + = with
i) x axis , y axis is1
ab2
Sq. units ii) y axis, z axis is1
bc2
Sq. units
iii) z axis, x axis is1
ca2
Sq. units
25. If a plane meets the coordinate axes in A,B,C such that the
centroid of the triangle ABC is the
point (p,q,r) then the equation of the plane is 3 x y z
p q r + + =
Angle between Two Planes:
26. The angle between two planes is equal to the angle between
the perpendiculars from the origin to
the planes.
27. If ' ' is the angle between the planes 1 1 1 1 0a x b y c z
d + + + = and 2 2 2 2 0a x b y c z d + + + = then
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
cosa a b b c c
a b c a b c
+ +=
+ + + +
28. If the above two planes are parallel then 1 1 1
2 2 2
a b c
a b c= =
If the above two planes are perpendicular then 1 2 1 2 1 2 0a a
b b c c+ + =
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