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E. Angle Between Two Straight Lines
Ifθ is the acute angle between two lines, then where m_{1 }and m_{2} are the slopes of the two lines and are finite.
Notes :
(i) If the two lines are perpendicular to each other then m_{1}m_{2} = 1
(ii) Any line perpendicular to ax + by + c = 0 is of the form bx  ay + k = 0
(iii) If the two lines are parallel or coincident, then m_{1}= m_{2}
(iv) Any line parallel to ax + by + c = 0 is of the form ax + by + k = 0
(v) If any of the two lines is perpendicular to xaxis, then the slope of that line is infinite.
Let m_{1} = ∝ , or θ = 90° – a , where tan a = m_{2}
i.e. angle θ is the complimentary to the angle which the oblique line makes with the xaxis.
(vi) If lines are equally inclined to the coordinate axis then m_{1} + m_{2} = 0
Ex.11 Find the equation to the straight line which is perpendicular bisector of the line segment AB, where A, B are (a,b) and (a', b') respectively.
Sol. Equation of AB is y  b =
i.e. y (a'  a)  x (b'  b) = a'b  ab'.
Equation to the line perpendicular to AB is of the form (b'  b)y + (a'  a)x + k = 0 ....(1)
Since the midpoint of AB lies on (1),
Hence the required equation of the straight line is
(1) Equation of straight Lines passing through a given point and equally inclined to a given line :
Let the straight passing through the point (x_{1}, y_{1}) and make equal angles with the given straight line y = mx + c. If m is the slope of the required line and a is the angle which this line makes with the given line then
(2) The above expression for tana, given two values of m, say m_{A} and m_{B}.
The required equations of the lines through the point (x_{1}, y_{1}) and making equal angles a with the given line are
y  y_{1} = m_{A}(x  x_{1}), y  y_{1} = m_{B} (x  x_{1})
Ex.12 Find the equation to the sides of an isosceles rightangled triangled, the equation of whose hypotenuse is 3x + 4y = 4 and the opposite vertex is the point (2, 2).
Sol. The problem can be restarted as :
Find the equation to the straight lines passing through the given point (2, 2) and making equal angles of 45° with the given straight line 3x + 4y  4 = 0. Slope of the line 3x + 4y  4 = 0 is
⇒ ,
i.e.,
m_{A} = , and m_{B} =  7
Hence the required equations of the two lines are
y  2 = m_{A}(x  2) and y  2 = m_{B} (x  2) ⇒ 7y  x  12 = 0 and 7x + y = 16.
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156 videos176 docs132 tests
