Quantitative Aptitude > Coordinate Geometry

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Coordinate Geometry

  • Two real number lines that are perpendicular to each other and that intersect at their respective zero points define a rectangular coordinate system, often called the xy-coordinate system or xy-plane. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. The point where the two axes intersect is called the origin, denoted by O. The positive half of the x-axis is to the right of the origin, and the positive half of the y-axis is above the origin. The two axes divide the plane into four regions called quadrants I, II, III, and IV, as shown in the figure below.


  • Each point P in the -plane can be identified with an ordered pair  of real numbers and is denoted by  The first number is called the -coordinate, and the second number is called the -coordinate. A point with coordinates  is located | units to the right of the -axis if  is positive or to the left of the -axis if x is negative. Also, the point is located | units above the -axis if  is positive or below the-axis if  is negative. If , the point lies on the -axis, and if , the point lies on the -axis. The origin has coordinates. Unless otherwise noted, the units used on the -axis and the-axis are the same.
  • In the figure above, the point P is  units to the right of the -axis and  units above the -axis, and the point  is  units to the left of the -axis and  units below the -axis.
  • Note that the three points  and  have the same coordinates as  except for the sign. These points are geometrically related to  as follows.
    • P′ is the reflection of P about the x-axis, or P′ and P are symmetric about the x-axis.
    • P″ is the reflection of P about the y-axis, or P″ and P are symmetric about the y-axis.
    • P′′′ is the reflection of P about the origin, or P′′′ and P are symmetric about the origin.
  • Equations in two variables can be represented as graphs in the coordinate plane. In the xy-plane, the graph of an equation in the variables x and y is the set of all points whose ordered pairs (x, y) satisfy the equation.
  • The graph of a linear equation of the form  is a straight line in the -plane, where  is called the slope of the line and  is called the -intercept.
  • The -intercepts of a graph are the -values of the points at which the graph intersects the -axis. Similarly, the -intercepts of a graph are the -values of the points at which the graph intersects the -axis.
  • The slope of a line passing through two points  and , where , is defined as

  • This ratio is often called "rise over run," where rise is the change in  when moving from Q to R and run is the change in when moving from Q to R. A horizontal line has a slope of 0, since the rise is 0 for any two points on the line. So the equation of every horizontal line has the form , where b is the -intercept. The slope of a vertical line is not defined, since the run is 0. The equation of every vertical line has the form , where  is the -intercept.
  • Two lines are parallel if their slopes are equal. Two lines are perpendicular if their slopes are negative reciprocals of each other. For e.g.  is perpendicular to the line with equation.

    Ex. In the -plane above, the slope of the line passing through the points  and  is

    Line QR appears to intersect the y-axis close to the point  so the -intercept of the line must be close to . To get the exact value of the -intercept, substitute the coordinates of any point on the line, say  into the equation , and solve it for  as follows.

    Therefore, the equation of line QR is .

    You can see from the graph that the -intercept of line QR is 2, since QR passes through the point . More generally, you can find the -intercept by setting  in an equation of the line and solving it for  as follows.

    Graphs of linear equations can be used to illustrate solutions of systems of linear equations and inequalities.

    Ex. Consider the following system of linear inequalities.


    Solving each inequality for  in terms of yields

    Each point (x, y) that satisfies the first inequality  is either on the line  or below the line because the y-coordinate is either equal to or less than . Therefore, the graph of  consists of the line  and the entire region below it. Similarly, the graph of  consists of the line  and the entire region above it. Thus, the solution set of the system of inequalities consists of all of the points that lie in the shaded region shown in the figure below, which is the intersection of the two regions described.



    Symmetry with respect to the x-axis, the y-axis, and the origin is mentioned above.

    Another important symmetry is symmetry with respect to the line with equation . The line  passes through the origin, has a slope of 1, and makes a 45-degree angle with each axis. For any point with coordinates , the point with interchanged coordinates  is the reflection of  about the line ; that is,  and  are symmetric about the line . It follows that interchanging  and  in the equation of any graph yields another graph that is the reflection of the original graph about the line .

    Ex.  Consider the line whose equation is . Interchanging  and  in the equation yields . Solving this equation for  yields . The line  and its reflection  are graphed below.

    The line  is a line of symmetry for the graphs of  and

  • The graph of a quadratic equation of the form , where  and  are constants and , is a parabola. The x-intercepts of the parabola are the solutions of the equation . If  is positive, the parabola opens upward and the vertex is its lowest point. If  is negative, the parabola opens downward and the vertex is the highest point. Every parabola is symmetric with itself about the vertical line that passes through its vertex. In particular, the two -intercepts are equidistant from this line of symmetry.

    Ex. The equation  has the following graph.


    The graph indicates that the -intercepts of the parabola are –1 and 3. The values of the -intercepts can be confirmed by solving the quadratic equation  to get  and . The point (1, –4) is the vertex of the parabola, and the line  is its line of symmetry. The -intercept is the -coordinate of the point on the parabola at which , which is .

    Ex. The graph of  is a circle with its center at the origin and with radius 10, as shown in the figure below. The smaller circle has center (6, –5) and radius 3, so its equation is .


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