1 Crore+ students have signed up on EduRev. Have you? 
The equation of the a straight lines bisecting the angles between the two straight lines ax^{2} + 2hxy + by^{2} = 0 is given by
The polar equation of a straight line is given by ... where p is the length of the perpendicular from the origin on the line
Figure is self explanatory. From A OMP
The polar equation of a line through two given points (r_{1}, θ_{1}) and (r_{2}, θ_{2}) is given by
A (r_{1}, θ) and B(r_{2}, θ_{2}) arc two given points through them passes the straight line L. Let P(r, θ) be an arbitrary point on L.
Then the points P, A and B are collinear.
=>the area of the triangle formed by these three points is zero
If parallelograms are described on the sides of a given triangle treating them as diagonals of t he respective parallelograms having their sides parallel to two given straight lines, then the other diagonals of these parallelograms
The other diagonal will always meet in a point.
If the equations of a pair of opposite sides of a parallelogram are given by x^{2}  7x + 6 = 0 and y^{2} 14y + 40 = 0, then the equation to one of it s diagonals is given by
x^{2} 7x + 6  0 => ( x  1) ( x 6)  0
⇒ x= 1 and x = 6 are two opposite sides
=> y=4 and y = 10 are two opposite sides
Clearly the four vertices of the parallelogram are
The equation ax^{2} + by^{2} + c(x + y) = 0 represents a pair of straight lines if
The equation of the straight line which passes through the point (x' ,y') and is perpendicular to the straight line yy' = 2a[x + x']is given by
The given straight line is yy' = 2a(x + x') ...(i)
the slope m of a straight line which ii perpendicular to (i) is given by
Now the required equation passes through (x',y') and posseses thes slope
∴ Its equation will be
The equation of the straight line passing through (x', y') and perpendicular to the straight line
The equations of straight lines passing through (x',y') and making an angle a with the given straight line y = mx + c, are given by
Let m' be the slope of the straight line which makes an angle a with the line
The equation of one of the straight lines passing through the point (h, k) and making an angle tan^{1} m with the. straight line y = mx + c is given ty
Proof: Let m' be the slope of the straight line which makes an angle tan^{1} m with the straight line y = mx + c. Then
∴ The straight line with slope m_{1} = 0 will be given by
The distance of the plane 6x 3y + 2z  14 = 0 from the origin is
The length of the perpendicular from (x_{1}, y_{1} ,z_{1}) to the plane Ax+ By  Cz + D = 0
is given by
∴ perpendicular distance d of the plane
6x3y2z 14=0
from the origin is given by
In the equation of the plane given by ax + by + cz + d = 0: a. b, c denote the
Note that an equation of first degree in x, y, z of the form
Ax + By + Cz + D = 0 ...(i)
represents a plane
d.r.’s of the normal to (i) are A,B,C.
∴ d.e.'s of the normal to (i) are
The angle between the two planes is same as the angle between
The angle between the two planes ax + by + cz + d = 0 and a'x + b'y + c'z  a' = 0 is given by
Proof: The given planes are
ax + by + cz d = 0
and a'x + b'y + c'z + d' = 0 The d.r.’s of their normals are
a, b, c and a', b', c' repsectively.
∴ The d.c.’s of their normals are
Therefore if 0 is the required angle between the two planes (and hence between their normals), then
That is why statements (a), (b) and (c) are not correct.
The two planes ax + by +by + cz + d = 0 and a'x + b'y + c'z + d = 0 are perpendicular if
Proof: The two planes are perpendicular if
θ = 90° or cos θ = 0
or aa' + bb' + cc' = 0
This is the condition of perpendicularity.
Remark : Condition of parallelism. Two planes are prallel if
The planes 2x  y + z = 15 and x + y + 2x = 3 are inclined at an angle of
The angle of inclination θ is given by
Which one of the following is incorrect? The condition that the four points (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3,} z_{3}) and (x_{4}, y_{4}, z_{4}) are coplanar is
The condition that the four points (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3}) and (x4, y4, z4) are coplanar is that
Remark: The other three statements are same as he condition given above, (rows or columns in a determinant can be interchanged. Even number of changes do not change the value of the determinant. Odd number of changes make the value of the determinant as the negative of the original value)
Which of the following statements is incorrect? two points P(x_{1} , y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) lie on the same side of the plane ax  by + cz  d 0 if
Result: Two points P(x_{1} , y_{1}, z_{1}) and Q (x_{2} , y_{2}, z_{2}) lie: on the same side: or opposite sides of the: plane
Ax + By + Cz + D = 0 according as the quantities
Ax_{1} + By_{1} + Cz_{1} + D and
Ax_{2} + By_{2} + Cz_{2} + D are of same or opposite signs.
The two points (1, 1, 1) and (3, 0, 1) from the plane 3x +4y  12z + 13=0 are
Proof: Substituting the coordinates of the given points P( 1,1, 1) and Q (3, 0, 1) in the equation of plane, we get
Since they are of opposite signs, therefore the points lie on the opposite sides of the plane. Also the perpendicular distances p and q of P and Q are given by
Thus the points are equidistant from the plane and on the opposite sides of it.
If D > 0, then the length of the perpendicular from the origin on the plane Ax + By + Cz + D = 0 is
Length of perpendicular from origin
27 docs150 tests

Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 
27 docs150 tests









