9. Straight lines (1M + 4M + 8M = 13M)
LONG ANSWER QUESTIONS ( 8 Marks ) :
1. Find the equation of the straight line passing through (−4, 5) and cutting off equal
and non-zero intercepts on the coordinate axes. [𝑩𝑶𝑨𝑹𝑫 𝑴𝑶𝑫𝑬𝑳 𝑷𝑨𝑷𝑬𝑹 − 𝟐]
2. Show that the straight lines (𝑎 − 𝑏)𝑥 + (𝑏 − 𝑐 )𝑦 = 𝑐 − 𝑎, (𝑏 − 𝑐 )𝑥 + (𝑐 − 𝑎)𝑦 =
𝑎 − 𝑏 𝑎𝑛𝑑 (𝑐 − 𝑎)𝑥 + (𝑎 − 𝑏)𝑦 = 𝑏 − 𝑐 are concurrent.
3. Two lines passing through the point (2, 3) intersects each other at and angle of 60°.
If slope of one line is 2, find equation of the other line.
4. If p and q are the lengths of perpendiculars from the origin to the lines
𝑥 cos 𝜃 − 𝑦 sin 𝜃 = 𝑘 cos 2𝜃 and 𝑥 sec 𝜃 + 𝑦 cosec 𝜃 = 𝑘 respectively, prove that
𝑝2 + 4 𝑞2 = 𝑘 2 .
5. A straight line meets the coordinate axes in A and B. Find the equation of the straight
line when i) ̅̅̅̅
AB is divide in the ratio 2 : 3 at (−5, 2).
ii) ̅̅̅̅
AB is divide in the ratio 1 : 2 at (−5, 4), iii) (𝑝, 𝑞) bisects ̅̅̅̅
AB.
6. Find the point on the straight line 3𝑥 + 𝑦 + 4 = 0 which is equidistant from the
points (−5, 6) and (3, 2).
𝜋
7. A straight line through Q (√3, 2) makes an angle with the positive direction of the x –
6
x - axis. If the straight line intersects the line √3 𝑥 − 4 𝑦 + 8 = 0 at P, find the distance PQ.
3𝜋
8. A straight line through Q (2, 3) makes an angle with the negative direction of the
4
x-axis. If the straight line intersects the line 𝑥 + 𝑦 − 7 = 0 at P, find the distance PQ.
9. Show that the straight lines 𝑥 + 𝑦 = 0, 3𝑥 + 𝑦 − 4 = 0 and 𝑥 + 3𝑦 − 4 = 0 from an
isosceies triangle.
10. Show that the area of the triangle formed by the line 𝑦 = 𝑚1 𝑥 + 𝑐1 , 𝑦 = 𝑚2 𝑥 + 𝑐2
(𝑐1 −𝑐2 )2
and 𝑥 = 0 is .
2|𝑚1 −𝑚2 |
11. A line is such that its segment between the lines 5𝑥 − 𝑦 + 4 = 0 𝑎𝑛𝑑 3𝑥 + 4𝑦 −
4 = 0 is bisected at the point (1, 5). Obtain its equation.
12. Show that the path of a moving point such that its distances from two lines
3𝑥 − 2𝑦 = 5 𝑎𝑛𝑑 3𝑥 + 2𝑦 = 5 are equal is a straight line.
13. Find the equation of the lines through the point (3, 2) which make an angle of 45°
with the line 𝑥 − 2𝑦 = 3.
14. Find the distance of the line 4𝑥 + 7𝑦 + 5 = 0 from the point (1, 2) along the line
2𝑥 − 𝑦 = 0.
, 15. Find the direction in which a straight line must be drawn through the point (−1, 2)so
that its point of intersection with the line 𝑥 + 𝑦 = 4 may be at a distance of 3 units
from this point.
16. If sum of the perpendicular distance of a variable point P(𝑥, 𝑦) from the lines
𝑥 + 𝑦 − 5 = 0 𝑎𝑛𝑑 3𝑥 − 2𝑦 + 7 = 0 is always10. show that P must move on a
line.
17. A ray of light passing through the point (1, 2) reflects on the x - axis at point A and
the reflected ray passes through the point (5, 3). Find the coordinates of A.
18. Prove that the product of the lengths of the perpendiculars drawn from the points
𝑥 𝑦
(√𝑎2 − 𝑏2 , 0) and (−√𝑎2 − 𝑏2 , 0) to the line cos 𝜃 + sin 𝜃 = 1 is 𝑏2 .
𝑎 𝑏
19. Show that the line 𝑥 − 7𝑦 − 22 = 0, 3𝑥 + 4𝑦 + 9 = 0 and 7𝑥 + 𝑦 − 54 = 0 form
a right angle isosceles triangle.
20. Find the equation of the straight lines passing through the point (−3, 2) and making
an angle of 45° with the straight line 3𝑥 − 𝑦 + 4 = 0.
SHORT ANSWER QUESTIONS ( 4 Marks ) :
𝜋 1
1. If the angle between two lines is 4 and slope of one of the lines is 2 , find the slope of
the other line.
2. Line through the points (−2, 6) and (4, 8) is perpendicular to the line through the
points (8, 12) and (𝑥, 24). Find the value of 𝑥.
3. Find a point on the x-axis, whch is equidistant from the point (7, 6) and (3, 4).
4. Without using the Pythagoras theorem, show the points (4, 4), (3, 5) and (−1, −1)
are the vertices of a right angled triangle.
5. Find the slope of the line, which makes an angle of 30° with the positive direction of
y-axis measured anti-clock wise.
6. Without using distance formula, show that points (−2, −1), (4, 0), (3, 3) and (−3, 2)
are the vertices of a parallelogram.
7. Find the angle between the x-axis and the line joing the points (3, −1) and (4, −2).
8. The slope of a line is double of the slope of another line. if tangent of the angle
1
between them is 3, find the slopes of the lines.
9. The vertices of ∆PQR are P(2, 1), Q(−2, 3) and R(4, 5). Find equation of the median
through the vertex R.
10. Find the equation of the line passing through (−3, 5) and perpendicular to the line
through the points (2, 5) and (−3, 6).
LONG ANSWER QUESTIONS ( 8 Marks ) :
1. Find the equation of the straight line passing through (−4, 5) and cutting off equal
and non-zero intercepts on the coordinate axes. [𝑩𝑶𝑨𝑹𝑫 𝑴𝑶𝑫𝑬𝑳 𝑷𝑨𝑷𝑬𝑹 − 𝟐]
2. Show that the straight lines (𝑎 − 𝑏)𝑥 + (𝑏 − 𝑐 )𝑦 = 𝑐 − 𝑎, (𝑏 − 𝑐 )𝑥 + (𝑐 − 𝑎)𝑦 =
𝑎 − 𝑏 𝑎𝑛𝑑 (𝑐 − 𝑎)𝑥 + (𝑎 − 𝑏)𝑦 = 𝑏 − 𝑐 are concurrent.
3. Two lines passing through the point (2, 3) intersects each other at and angle of 60°.
If slope of one line is 2, find equation of the other line.
4. If p and q are the lengths of perpendiculars from the origin to the lines
𝑥 cos 𝜃 − 𝑦 sin 𝜃 = 𝑘 cos 2𝜃 and 𝑥 sec 𝜃 + 𝑦 cosec 𝜃 = 𝑘 respectively, prove that
𝑝2 + 4 𝑞2 = 𝑘 2 .
5. A straight line meets the coordinate axes in A and B. Find the equation of the straight
line when i) ̅̅̅̅
AB is divide in the ratio 2 : 3 at (−5, 2).
ii) ̅̅̅̅
AB is divide in the ratio 1 : 2 at (−5, 4), iii) (𝑝, 𝑞) bisects ̅̅̅̅
AB.
6. Find the point on the straight line 3𝑥 + 𝑦 + 4 = 0 which is equidistant from the
points (−5, 6) and (3, 2).
𝜋
7. A straight line through Q (√3, 2) makes an angle with the positive direction of the x –
6
x - axis. If the straight line intersects the line √3 𝑥 − 4 𝑦 + 8 = 0 at P, find the distance PQ.
3𝜋
8. A straight line through Q (2, 3) makes an angle with the negative direction of the
4
x-axis. If the straight line intersects the line 𝑥 + 𝑦 − 7 = 0 at P, find the distance PQ.
9. Show that the straight lines 𝑥 + 𝑦 = 0, 3𝑥 + 𝑦 − 4 = 0 and 𝑥 + 3𝑦 − 4 = 0 from an
isosceies triangle.
10. Show that the area of the triangle formed by the line 𝑦 = 𝑚1 𝑥 + 𝑐1 , 𝑦 = 𝑚2 𝑥 + 𝑐2
(𝑐1 −𝑐2 )2
and 𝑥 = 0 is .
2|𝑚1 −𝑚2 |
11. A line is such that its segment between the lines 5𝑥 − 𝑦 + 4 = 0 𝑎𝑛𝑑 3𝑥 + 4𝑦 −
4 = 0 is bisected at the point (1, 5). Obtain its equation.
12. Show that the path of a moving point such that its distances from two lines
3𝑥 − 2𝑦 = 5 𝑎𝑛𝑑 3𝑥 + 2𝑦 = 5 are equal is a straight line.
13. Find the equation of the lines through the point (3, 2) which make an angle of 45°
with the line 𝑥 − 2𝑦 = 3.
14. Find the distance of the line 4𝑥 + 7𝑦 + 5 = 0 from the point (1, 2) along the line
2𝑥 − 𝑦 = 0.
, 15. Find the direction in which a straight line must be drawn through the point (−1, 2)so
that its point of intersection with the line 𝑥 + 𝑦 = 4 may be at a distance of 3 units
from this point.
16. If sum of the perpendicular distance of a variable point P(𝑥, 𝑦) from the lines
𝑥 + 𝑦 − 5 = 0 𝑎𝑛𝑑 3𝑥 − 2𝑦 + 7 = 0 is always10. show that P must move on a
line.
17. A ray of light passing through the point (1, 2) reflects on the x - axis at point A and
the reflected ray passes through the point (5, 3). Find the coordinates of A.
18. Prove that the product of the lengths of the perpendiculars drawn from the points
𝑥 𝑦
(√𝑎2 − 𝑏2 , 0) and (−√𝑎2 − 𝑏2 , 0) to the line cos 𝜃 + sin 𝜃 = 1 is 𝑏2 .
𝑎 𝑏
19. Show that the line 𝑥 − 7𝑦 − 22 = 0, 3𝑥 + 4𝑦 + 9 = 0 and 7𝑥 + 𝑦 − 54 = 0 form
a right angle isosceles triangle.
20. Find the equation of the straight lines passing through the point (−3, 2) and making
an angle of 45° with the straight line 3𝑥 − 𝑦 + 4 = 0.
SHORT ANSWER QUESTIONS ( 4 Marks ) :
𝜋 1
1. If the angle between two lines is 4 and slope of one of the lines is 2 , find the slope of
the other line.
2. Line through the points (−2, 6) and (4, 8) is perpendicular to the line through the
points (8, 12) and (𝑥, 24). Find the value of 𝑥.
3. Find a point on the x-axis, whch is equidistant from the point (7, 6) and (3, 4).
4. Without using the Pythagoras theorem, show the points (4, 4), (3, 5) and (−1, −1)
are the vertices of a right angled triangle.
5. Find the slope of the line, which makes an angle of 30° with the positive direction of
y-axis measured anti-clock wise.
6. Without using distance formula, show that points (−2, −1), (4, 0), (3, 3) and (−3, 2)
are the vertices of a parallelogram.
7. Find the angle between the x-axis and the line joing the points (3, −1) and (4, −2).
8. The slope of a line is double of the slope of another line. if tangent of the angle
1
between them is 3, find the slopes of the lines.
9. The vertices of ∆PQR are P(2, 1), Q(−2, 3) and R(4, 5). Find equation of the median
through the vertex R.
10. Find the equation of the line passing through (−3, 5) and perpendicular to the line
through the points (2, 5) and (−3, 6).
