Question 1 :
If the straight lines
y/2 = x – p and ax + 5 = 3y
are parallel, then find a.
Question 2 :
Find the value of a if the straight lines
5x – 2y – 9 = 0 and ay + 2x – 11 = 0
are perpendicular to each other.
Question 3 :
Find the value of p for which the straight lines
8px + (2 - 3p)y + 1 = 0 and px + 8y – 7 = 0
are perpendicular to each other.
Question 4 :
If the straight line passing through the points
(h, 3) and (4, 1)
intersects the line
7x – 9y – 19 = 0
at a right angle, find the value of h.
1. Answer :
y/2 = x – p and ax + 5 = 3y
Since the given lines are parallel, they will have same slope.
Slope of the line y/2 = x – p :
y/2 = x – p
y = 2(x - p)
y = 2x – 2p
Comparing y = 2x – 2p and y = mx + b, we get
m = 2
Slope of the 1^{st} line (m_{1}) = 2 ----(1)
Slope of the line 3 y = ax + 5 :
3y = ax + 5
y = (ax + 5)/3
y = (a/3)x + (5/3)
Slope of the 2^{nd} line (m_{2}) = a/3 ----(2)
(1) = (2)
2 = a/3
a = 6
2. Answer :
5x – 2y – 9 = 0 and ay + 2x – 11 = 0
If two lines are perpendicular then,
slope of the 1^{st} line x slope of the 2^{nd} line = -1
Slope of the line 5x – 2y – 9 = 0 :
5x – 2y – 9 = 0
slope (m) = -coefficient of x/coefficient of y
m_{1} = -5/(-2)
m_{1 }= 5/2 ----(1)
Slope of the line ay + 2x – 11 = 0 :
2x + ay - 11 = 0
m_{2} = -2/a ----(2)
Product of slopes of two lines = -1
(5/2) x (-a/2) = -1
-5a/4 = -1
-5a = -4
a = -4/(-5)
a = 4/5
3. Answer :
8px + (2 - 3p)y + 1 = 0 and px + 8y – 7 = 0
If two lines are perpendicular then,
slope of the 1^{st} line x slope of the 2^{nd} line = -1
Slope of the line 8px+(2-3p)y+1 = 0 :
m_{1} = -8p/(2 - 3p) ----(1)
Slope of the line px + 8y – 7 = 0 :
m_{2} = -p/8 ----(2)
m_{1} x m_{2} = -1
-8p/(2 - 3p) x (-p/8) = -1
p^{2}/(2 - 3p) = -1
p^{2} = -1(2 - 3p)
p^{2 }= -2 + 3p
p^{2 }- 3p + 2 = 0
(p - 1)(p - 2) = 0
p – 1 = 0 or p – 2 = 0
p = 1 or p = 2
4. Answer :
Here the straight which is passing through the points (h, 3) and (4, 1) is perpendicular to the line 7x – 9y – 19 = 0.
Slope of the line when two points are given :
m = (y_{2 }- y_{1})/(x_{2 }- x_{1})
m_{1} = (1 - 3)/(4 - h)
m_{1} = -2/(4 - h) ----(1)
m_{2} = -7/(-9)
m_{2 }= 7/9 ----(2)
[-2/(4 - h)] x (7/9) = -1
[2/(4 - h)] x [7/9] = 1
14/9(4 - h) = 1
14 = 36 – 9h
9h = 36 – 14
9h = 22
h = 22/9
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