# Second Year Math Introduction to Analytic Geometry 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 lin

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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):8. Find a joint equation of the lines through the origin and perpendicular to the lines:\[a x^{2}+2 h x y+b y^{2}=0\]

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)\[\begin{array}{l}l_{1}: \text { Joining }(27) \text { and }(710) \\l_{2}: \text { Joining }(11) \text { and }(-53)\end{array}\]

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) .

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.

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)

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

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)

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.

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)

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\[\begin{array}{l}l: 2 x-5 y+13=0 \text { and } \\l_{2}: 2 x-5 y+6=0\end{array}\]

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.

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\[\text { (d) } A(28) B(-54) C(4-9)\]

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

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)

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.