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Question (1):
Find the value of 'a' for which the lines ax + 4y + 5 = 0 and 9x + ay - 4 = 0 are parallel to each other.
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Answer:
ax + 4y + 5 = 0 and 9x + ay - 4 = 0 are parallel to each other, the slopes are equal.
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Question (2):
Find the equation of the line perpendicular to the line 8x + 5y = 7 and passing through (1, 2).
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Answer:
( Product of slopes of two perpendicular lines = -1)
Aliter:
Given line is 8x + 5y = 7.
The line perpendicular to 8x + 5y = 7 is 5x - 8y = k.
This line passes through (1,2).
 The required equation of the line is
5x - 8y = -11 or 5x - 8y + 11 = 0
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Question (3):
Find the equation of the line, which bisects the line joining (3, -1) and (5,11) and also bisects the distance between the points (-5,2) and (9,6).
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Answer:
The co-ordinates of the middle point of the line joining (3,-1) and (5,11) is
Let P (4,5) be the point.
The co-ordinates of the middle point of the line joining (-5,2) and (9, 6) is
Let Q (2,4) be the point.
We have to find the equation of the line joining the points P and Q.
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Question (4):
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Answer:
\ The required equation is
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Question (5):
Find the image of the point (-1,1) w.r.t to the line 3x + 4y = 32.
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Answer:
Let the image of the point P (-1,1) about the line AB be Q (h,k).
 PQ is perpendicular to AB and is bisected at C.
Solving (ii) and (i) for h and k, we have
 Reflection of P (-1,1) about the line 3x + 4y = 32 is
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Question (6):
Find the co-ordinates of the foot of the perpendicular from the point (4, -1) on the line x + 4y = 2.
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Answer:
Slope of line perpendicular to x + 4y = 2 is 4.
The point of intersection of the line given the co-ordinates of the foot of perpendicular is
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Question (7):
Find the equation of two lines passing through the point (4,5) 
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Answer:
The equation of the line through (4,5) with slope m is
y - 5 = m(x - 4)
The slope of line 2x - y + 7 = 0 is 2.
 The required lines are
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Question (8):
Find the equation of the line, which makes intercepts of 2 and 3 with x-axis and y-axis respectively.
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Answer:
Here,
a = 2, b = 3
The required equation of the line is
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Question (9):
Find the equation of the straight line which passes through (4,5) and sum of intercepts on the coordinate axes is 18.
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Answer:
Let the intercept on the x-axis be a, then the intercept on the y-axis will be (18 - a).
The line passes through (4,5).
a = 8 or a = 9
When a = 8, b = 9 or when a = 9, b = 8
 The required equations are
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Question (10):
Find the equation of the line through (a,b) and the sum of the intercepts on the coordinate axes is 2(a+b).
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Answer:
Let s be the intercept on the x-axis, then the intercept on y-axis will be
2(a + b) - s.
Let t = 2(a+b)-s
This passes through (a,b).
Equation of the line is
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Question (11):
If p is the length of perpendicular drawn from the origin on the line AB whose intercepts on the axes are a and b, then show 
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Answer:
Aliter:
Equation of the line AB in the intercept form is
 ...(i)
Since (i) and (ii) represent the same line, we have
Squaring and adding, we get
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Question (12):
A straight line passes through the point (a, b) and this bisects the part of the line intercepted between the axes.
Show that 
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Answer:
Let the straight line passing through p(a, b) cut the x-axis at A(a,0) and B(0,b) on the x-axis and the y-axis respectively.
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Question (13):
Find the equation of the straight line which passes through the point (2,3) and whose intercepts on the y-axis is thrice that on the x-axis.
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Answer:
Let the equation of the line in the intercept form be
From the problem, b = 3a.
(ii) passes through (2,3).
b = 3a = 3 x 3 = 9
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Question (14):
Find the equation of the straight line which passes through the point (3,2) and cuts off intercepts a and b respectively on the x and y axis such that a - b = 2.
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Answer:
Let the equation of the line be
 ...(i)
From the problem, a - b = 2
a = b+2
This line passes through (3,2).
 The two lines are
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Question (15):
Find the equation of the line which makes thrice the intercepts made by the line 3x + 4y = 12 on the coordinate axes.
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Answer:
The given line is 3x+4y = 12. ......(i)
The intercept made by the line on the x-axis is obtained by substituting
y = 0 in (i).
The intercept made by the line on the y-axis is obtained by substituting
x = 0 in (i).
The intercepts made by the required line are 3a and 3b.
i.e. 12 and 9 respectively.
 Equation of the required line is
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Question (16):
Find the equation of the straight line, which passes through p(1,-7) and meets the axes at A and B respectively, so that 4AP - 3BP = 0, where O is the origin.
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Answer:
Now, the point p(1,-7) divides AB in the ratio AP:PB::3:4.
Let the coordinates of A be (a,0) and that of B be (0,b).
 The equation of the line is
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Question (17):
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Answer:
Let C divide AB in the ratio m:n. The coordinates of C are
Since lies on ax + by + c = 0, we have
which is the required ratio.
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Question (18):
Find the equations of the diagonals of the quadrilateral whose  ratio in which each diagonal divides the other.
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Answer:
Equation of diagonal AC is
or
Let this line divide DB in the ratio m:n.
Equation of the diagonal BD is
Let this line divide the diagonal AC in the ratio m:n.
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Question (19):
Find the equation of the line passing through (2,-1) and making an angle of 45o with the positive direction of the x-aixs. Also find the points on the line at a distance from the point (2,-1).
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Answer:
Here,
Equation of the line is
y + 1 = 1(x - 2)
 x - y = 3
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Question (20):
Find the equation of the line which passes through the point
 points on the line that is 5 units away from the point (3,2).
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Answer:
The equation of the line in the point-slope form is
Here,
 The points on the line at a distance 5 units from (3,2).
 The points are (7,5) and (-1,-1).
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