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The equation of a straight line which makes an angle of 60° with xaxis and passes through the point (√3, 2) is given by
The required equation
(i) makes an angle of 60° with xaxis
(ii) passes through the point (√3 , 2)
Now
(i) ⇒ equation of the straight line will be
y = tan 60° x + c
or y = √3 x + c ...(i)
(ii) ⇒ the coordinates of the point will satisfy equation (i)
The line y  x + 2 = 0 cuts the line joining (3, 1) and (8, 9) in the ratio
First remember the following points:
l.The equation of a line passing through two p o in ts (x_{1},y_{1}) and (x_{2}, y_{2}) is given by
2. The. coordinates of the point R which divides the line joining P(x_{1},y_{1}) and Q(x_{2}, y_{2}) internally in the ratio m_{1} : m_{2} are given by
3. If R divides PQ externally in the ratio m_{1} : m_{2}, then
Now the given straight line is
y  x + 2  0 ...(iv)
The equation of the straight line joining the points (3,1) and (8, 9) is given by (see (i))
or y  2x + 7 = 0 ...(v)
The point of intersection of (iv) and (v) is obtained on solving these equations and is given by (5,3)
Let the points of intersection (5, 3) divide the line joining (3. 4) and (8. 9) in the ratio m : n.
Then
The equation of the straight line passing through (4, 5) and parallel to the line 2x  3y = 5 is given by
The given straight line is
The equation of a straight line parallel to (i) will be
Since (ii) passes through P(4, 5), therefore the coordinates of P A\*ill satisfy equation (ii)
∴ Required equation of the straight line is
The condition that the three straight lines
ax + by+ c = 0
a_{1}x + b_{1}y + c = 0
a_{2}x + b_{2}y + c_{2} = 0
may meet in a point, is that, the value of the determinant must be
The condition for three lines lines to be concurrent is that
The point (x_{1},y_{1}) lies on the positive side of the straight line Ax + By + C = 0
First of all learn the following:
Division of a plane by the line
L : Ax + By + C = 0
As shown in the adjoining Figure, the line L divides the plane into three parts, namely I, L and II, where
Ax_{1} + By_{1} + C > 0 for all poins (x_{1}, y_{1}) in l
Ax_{2} + By_{2} + C = 0 for all points (x_{2}, y_{2}) on L
Ax_{3} + By_{3} + C < 0 for all points (x_{3}, y_{3}) in II
Given point is (3, 2)
Given straight lines are
When two straight lines 2x 3y + 1 = 0 and 3x 6y + 2 = 0 are traced, we get four different compartments. Which of the following four points lie in the same compartment? A(0, 0), B(1, 1), C(7, 4), D(9, 6)
The given straight lines are
The angle between the lines ax + by + c = 0 and a'x + b'x + c' = 0 is given by
Comments. The angle φ between two straight lines
y = mx + c and y = m'x + c‘
is given by
Since the straight lines are ax + by + c = 0 and a'x +b'y + c' = 0
therefore
Both φ and π  φ shall be the angles between the given lines.
The straight lines ax + by + c = 0 and a'x + b'y + c' = 0 arc perpendicular if
The angle φ between the lines is given by
Remark: A straight line passing through the origin shall be of the form
y = mx
and there will not bo any constant term in the eouation.
The four points (0, 4, 1), (2, 3, 1), (4, 5, 0). (2, 6, 2) are the vertices of a
Let A(3, 2, 0), B(5, 3, 2), C( 9, 6, 3) be the three points forming a triangle. Let the bisector of the angle BAC meet BC in D. Then D divides BC in the ratio of
Let A(1,2, 3), B(5, 0, 6), C(0, 4, 1) be the three points. Then the direction cosines of the internal bisector of the angle BAC are proportional to
Proof in short : verify that
AB = 7 and AC = 3
where θ is the angle between AB (with d.c.'s [l_{1}, m_{1}, n_{1}]) and AC (with d.c.’s [l_{2}, m_{2}, n_{2})
∴ the d.c.’s of the internal bisector are proprotional to l_{1}+l_{2}, m_{1}+m_{2}, n_{1}+n_{2}
or proportional to
or proportional to
or proportional to 25, 8, 5
Which of the following is incorrect?
If l_{1}, m_{1}, n_{1}, : l_{2}, m_{2}, n_{2}, : l_{3}, m_{3}, n_{3} be the direction cosines of three mutually perpendicular lines then
In fact, we shall have
The direction ratios of the line, which is equally inclined to the three mutually perpendicular lines with direction cosines l_{1}, m_{1}, n_{1}: l_{2}, m_{2}, n_{2} : l_{3}, m_{3}, n_{3}; are given by
Proof: Let the three mutually perpendicular lines be
“If O, A, B, C be the four points not lying in the same plane such that OA ⊥ BC, OB ⊥ CA, then OC ⊥ AB". if the points O, A, B, C are coplanar, then the above reduces to
If a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are the direction ratios of two lines, then the angle between them is not given by
If the edges of a rectangular parallelepiped are OA , OB and OC along coordinate axes such that OA = a, OB = b, OC = c and PL, PM and PN are the perpendiculars drawn from P(a, b, c) on XY, ZX and YZ planes respectively, then angle between OP and AN is given by
If a variable line in two adjacent positions has direction cosines as then the small angle dθ between those two positions.in given by
The area of the triangle with vertices (0, 0, 0), (0, b, 0) and (0, 0, c) is given by
Proof: The area of the triangle OAB, where O is the origin and the coordinates of A and B are (x_{1}, y_{1}, z_{1}) a n d [x_{2}, y_{2}, z_{2}) respectively,
Here since the coordinates of A and B are (0,b,0) and (0.0,c), therefore the required area A is given by
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