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Second Year Math Introduction to Analytic Geometry 28. Find whether the given point lies above or below the given line(a) (58) ; 2 x-3 y+6=0 (b) (-76) ; 4 x+3 y-9=0


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28. Find whether the given point lies above or below the given line(a) (58) ; 2 x-3 y+6=0 (b) (-76) ; 4 x+3 y-9=0

23. Find the distance between the given parallel lines. Sketch the lines. Also find an equation of the parallel line lying midway between them.(a) 3 x-4 y+3=0 ; 3 x-4 y+7=0
23. Find the distance between the given parallel lines. Sketch the lines. Also find an equation of the parallel line lying midway between them.(a)  3 x-4 y+3=0  ;  3 x-4 y+7=0

23. Find the distance between the given parallel lines. Sketch the lines. Also find an equation of the parallel line lying midway between them.(a) 3 x-4 y+3=0 ; 3 x-4 y+7=0

10. Find the angle measured from the line l_{1} to the line l_{2} where(a)\[\begin{array}{l}l_{1}: \text { Joining }(27) \text { and }(710) \\l_{2}: \text { Joining }(11) \text { and }(-53)\end{array}\]
10. Find the angle measured from the line  l_{1}  to the line  l_{2}  where(a)\[\begin{array}{l}l_{1}: \text { Joining }(27) \text { and }(710) \\l_{2}: \text { Joining }(11) \text { and }(-53)\end{array}\]

10. Find the angle measured from the line l_{1} to the line l_{2} where(a)\[\begin{array}{l}l_{1}: \text { Joining }(27) \text { and }(710) \\l_{2}: \text { Joining }(11) \text { and }(-53)\end{array}\]

3. By means of slopes show that the following points lie on the same line:(a) (-1-3) ;(15) ;(29)
3. By means of slopes show that the following points lie on the same line:(a)  (-1-3) ;(15) ;(29)

3. By means of slopes show that the following points lie on the same line:(a) (-1-3) ;(15) ;(29)

14. Find the point three-fifth of the way along the line segment from A(-58) to B(53) .
14. Find the point three-fifth of the way along the line segment from  A(-58)  to  B(53) .

14. Find the point three-fifth of the way along the line segment from A(-58) to B(53) .

10. Find the angle measured from the line l_{1} to the line l_{2} where(b) l_{1}: Joining (3-1) and (57) l_{2}: Joining (24) and (-82)
10. Find the angle measured from the line  l_{1}  to the line  l_{2}  where(b)  l_{1}:  Joining  (3-1)  and  (57)  l_{2}:  Joining  (24)  and  (-82)

10. Find the angle measured from the line l_{1} to the line l_{2} where(b) l_{1}: Joining (3-1) and (57) l_{2}: Joining (24) and (-82)

Example 5: Find an equation of the line through the point P(23) which forms an isosceles triangle with the coordinate axes in the first rant.
Example 5:    Find an equation of the line through the point  P(23)  which forms an isosceles triangle with the coordinate axes in the first rant.

Example 5: Find an equation of the line through the point P(23) which forms an isosceles triangle with the coordinate axes in the first rant.

7. The vertices of a triangle are A(-23) B(-41) and C(35) . Find coordinates of the(iii) circumcentre of the triangle Are these three points collinear?
7. The vertices of a triangle are  A(-23) B(-41)  and  C(35) . Find coordinates of the(iii) circumcentre of the triangle Are these three points collinear?

7. The vertices of a triangle are A(-23) B(-41) and C(35) . Find coordinates of the(iii) circumcentre of the triangle Are these three points collinear?

2. The x y -coordinate axes are translated through the point whose coordinates are given in x y -coordinate system. The coordinates of P are given in the X Y -coordinate system. Find the coordinates of P in x y -coordinate system.(iv) P(4-3) ; 0^{\prime}(-23)
2. The  x y -coordinate axes are translated through the point whose coordinates are given in  x y -coordinate system. The coordinates of  P  are given in the  X Y -coordinate system. Find the coordinates of  P  in  x y -coordinate system.(iv)  P(4-3) ; 0^{\prime}(-23)

2. The x y -coordinate axes are translated through the point whose coordinates are given in x y -coordinate system. The coordinates of P are given in the X Y -coordinate system. Find the coordinates of P in x y -coordinate system.(iv) P(4-3) ; 0^{\prime}(-23)

16. Express the given system of equations in matrix form. Find in each case whether the lines are concurrent.(c) 3 x-4 y-2=0 ; x+2 y-4=0 ; 3 x-2 y+5=0 .
16. Express the given system of equations in matrix form. Find in each case whether the lines are concurrent.(c)  3 x-4 y-2=0 ;  x+2 y-4=0 ;  3 x-2 y+5=0 .

16. Express the given system of equations in matrix form. Find in each case whether the lines are concurrent.(c) 3 x-4 y-2=0 ; x+2 y-4=0 ; 3 x-2 y+5=0 .

28. Find whether the given point lies above or below the given line(a) (58) ; 2 x-3 y+6=0 (b) (-76) ; 4 x+3 y-9=0
28. Find whether the given point lies above or below the given line(a)  (58) ; 2 x-3 y+6=0 (b)  (-76) ;  4 x+3 y-9=0
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28. Find whether the given point lies above or below the given line(a) (58) ; 2 x-3 y+6=0 (b) (-76) ; 4 x+3 y-9=0

12. Find equations of the sides altitudes and medians of the triangle whose vertices are A(-32) B(54) and C(3-8) .
12. Find equations of the sides altitudes and medians of the triangle whose vertices are  A(-32) B(54)  and  C(3-8) .

12. Find equations of the sides altitudes and medians of the triangle whose vertices are A(-32) B(54) and C(3-8) .

1. The two points P and O^{\prime} are given in x y -coordinate system. Find the X Y -coordinates of P reffered to the translated axes O^{\prime} X and O^{\prime} Y .(iii) P(-6-8) ; O^{\prime}(-4-6)
1. The two points  P  and  O^{\prime}  are given in  x y -coordinate system. Find the  X Y -coordinates of  P  reffered to the translated axes  O^{\prime} X  and  O^{\prime} Y .(iii)  P(-6-8) ; O^{\prime}(-4-6)

1. The two points P and O^{\prime} are given in x y -coordinate system. Find the X Y -coordinates of P reffered to the translated axes O^{\prime} X and O^{\prime} Y .(iii) P(-6-8) ; O^{\prime}(-4-6)

Example 6: The length of perpendicular from the origin to a line is 5 units and the inclination of this perpendicular is 120^{\circ} . Find the slope and y -intercept of the line.
Example 6: The length of perpendicular from the origin to a line is 5 units and the inclination of this perpendicular is  120^{\circ} . Find the slope and  y -intercept of the line.

Example 6: The length of perpendicular from the origin to a line is 5 units and the inclination of this perpendicular is 120^{\circ} . Find the slope and y -intercept of the line.

Example 1: The coordinates of a point P are (-69) . The axes are translated through the point O^{\prime}(-32) . Find the coordinates of P referred to the new axes.
Example 1:    The coordinates of a point  P  are  (-69) . The axes are translated through the point  O^{\prime}(-32) . Find the coordinates of  P  referred to the new axes.

Example 1: The coordinates of a point P are (-69) . The axes are translated through the point O^{\prime}(-32) . Find the coordinates of P referred to the new axes.

Find the lines represented by each of the following and also find measure of the angle between them (Problems 1-6):8. Find a joint equation of the lines through the origin and perpendicular to the lines:\[a x^{2}+2 h x y+b y^{2}=0\]
Find the lines represented by each of the following and also find measure of the angle between them (Problems 1-6):8. Find a joint equation of the lines through the origin and perpendicular to the lines:\[a x^{2}+2 h x y+b y^{2}=0\]

Find the lines represented by each of the following and also find measure of the angle between them (Problems 1-6):8. Find a joint equation of the lines through the origin and perpendicular to the lines:\[a x^{2}+2 h x y+b y^{2}=0\]

7. The vertices of a triangle are A(-23) B(-41) and C(35) . Find coordinates of the (i) centroid
7. The vertices of a triangle are  A(-23) B(-41)  and  C(35) . Find coordinates of the (i) centroid

7. The vertices of a triangle are A(-23) B(-41) and C(35) . Find coordinates of the (i) centroid

11. Find an equation of the perpendicular bisector of the segment joining the points A(35) and B(98)
11. Find an equation of the perpendicular bisector of the segment joining the points  A(35)  and  B(98)

11. Find an equation of the perpendicular bisector of the segment joining the points A(35) and B(98)

6. Find h such that the points A(\sqrt{3}-1) B(02) and C(h-2) are vertices of a right
6. Find  h  such that the points  A(\sqrt{3}-1) B(02)  and  C(h-2)  are vertices of a right

6. Find h such that the points A(\sqrt{3}-1) B(02) and C(h-2) are vertices of a right

Example: Find the distance between the parallel lines\[\begin{array}{l}l: 2 x-5 y+13=0 \text { and } \\l_{2}: 2 x-5 y+6=0\end{array}\]
Example: Find the distance between the parallel lines\[\begin{array}{l}l: 2 x-5 y+13=0 \text { and } \\l_{2}: 2 x-5 y+6=0\end{array}\]
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Example: Find the distance between the parallel lines\[\begin{array}{l}l: 2 x-5 y+13=0 \text { and } \\l_{2}: 2 x-5 y+6=0\end{array}\]

3. Which of the following points are at a distance of 15 units from the origin?(a) (\sqrt{176} 7)
3. Which of the following points are at a distance of 15 units from the origin?(a)  (\sqrt{176} 7)
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3. Which of the following points are at a distance of 15 units from the origin?(a) (\sqrt{176} 7)

32. The coordinates of three points are A(23) B(-11) and C(4-5) . By computing the area bounded by A B C check whether the points are collinear.
32. The coordinates of three points are  A(23) B(-11)  and  C(4-5) . By computing the area bounded by  A B C  check whether the points are collinear.
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32. The coordinates of three points are A(23) B(-11) and C(4-5) . By computing the area bounded by A B C check whether the points are collinear.

Example 3: Find the distance between the parallel lines 2 x+y+2=0 6 x+3 y-8=0 (2)
Example 3:    Find the distance between the parallel lines  2 x+y+2=0   6 x+3 y-8=0 (2)
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Example 3: Find the distance between the parallel lines 2 x+y+2=0 6 x+3 y-8=0 (2)

11. Find the interior angles of the triangle whose vertices are\[\text { (d) } A(28) B(-54) C(4-9)\]
11. Find the interior angles of the triangle whose vertices are\[\text { (d) }  A(28)  B(-54) C(4-9)\]
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11. Find the interior angles of the triangle whose vertices are\[\text { (d) } A(28) B(-54) C(4-9)\]

16. Find the point which is equidistant from the points A (53) B(-22) and C(42) . What is the radius of the circumcircle of the \triangle \mathrm{ABC} ?
16. Find the point which is equidistant from the points  A (53)   B(-22)  and  C(42) . What is the radius of the circumcircle of the  \triangle \mathrm{ABC}  ?
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16. Find the point which is equidistant from the points A (53) B(-22) and C(42) . What is the radius of the circumcircle of the \triangle \mathrm{ABC} ?

Find the lines represented by each of the following and also find measure of the angle between them (Problems 1-6):4. 2 x^{2}+3 x y-5 y^{2}=0
Find the lines represented by each of the following and also find measure of the angle between them (Problems 1-6):4.   2 x^{2}+3 x y-5 y^{2}=0
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Find the lines represented by each of the following and also find measure of the angle between them (Problems 1-6):4. 2 x^{2}+3 x y-5 y^{2}=0

3. The x y -coordinate axes are rotated about the origin through the indicated angle. The new axes are O X and O Y . Find the X Y -coordinates of the point P with the given x y -coordinates.(ii) P(3-7) ; \theta=30^{\circ}
3. The  x y -coordinate axes are rotated about the origin through the indicated angle. The new axes are  O X  and  O Y . Find the  X Y -coordinates of the point  P  with the given  x y -coordinates.(ii)   P(3-7) ;  \theta=30^{\circ}
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3. The x y -coordinate axes are rotated about the origin through the indicated angle. The new axes are O X and O Y . Find the X Y -coordinates of the point P with the given x y -coordinates.(ii) P(3-7) ; \theta=30^{\circ}

1. The two points P and O^{\prime} are given in x y -coordinate system. Find the X Y -coordinates of P reffered to the translated axes O^{\prime} X and O^{\prime} Y .(ii) P(-26) ; O^{\prime}(-32)
1. The two points  P  and  O^{\prime}  are given in  x y -coordinate system. Find the  X Y -coordinates of  P  reffered to the translated axes  O^{\prime} X  and  O^{\prime} Y .(ii)  P(-26) ; O^{\prime}(-32)
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1. The two points P and O^{\prime} are given in x y -coordinate system. Find the X Y -coordinates of P reffered to the translated axes O^{\prime} X and O^{\prime} Y .(ii) P(-26) ; O^{\prime}(-32)

Example 1: Check whether the following lines are concurrent or not. If concurrent find the point of concurrency.\[\begin{array}{l}3 x-4 y-3=0 \\5 x+12 y+1=0 \\32 x+4 y-17=0\end{array}\]
Example 1:    Check whether the following lines are concurrent or not. If concurrent find the point of concurrency.\[\begin{array}{l}3 x-4 y-3=0 \\5 x+12 y+1=0 \\32 x+4 y-17=0\end{array}\]
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Example 1: Check whether the following lines are concurrent or not. If concurrent find the point of concurrency.\[\begin{array}{l}3 x-4 y-3=0 \\5 x+12 y+1=0 \\32 x+4 y-17=0\end{array}\]

4. Find the condition that the lines y=m_{1} x+c_{1} ; y=m_{2} x+c_{2} and y=m_{3} x+c_{3} are concurrent.
4. Find the condition that the lines  y=m_{1} x+c_{1} ; y=m_{2} x+c_{2}  and  y=m_{3} x+c_{3}  are concurrent.
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4. Find the condition that the lines y=m_{1} x+c_{1} ; y=m_{2} x+c_{2} and y=m_{3} x+c_{3} are concurrent.

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