Descriptive Geometry

The descriptive geometry is a branch of geometry, which deals three dimensional objects in two dimensions by using the lines, curves, solids, surfaces and points in space. Geo means “earth” and metron  means “measurement”. ”Euclid, is a Greek mathematician, called the father of geometry. A point is used to represent a position in space. A plane to be a surface extending infinitely in every directions such that all points lying on the line joining any two points on the surface. The descriptive geometry example problems and practice problems are given below. 

Example problems for descriptive geometry:

Example problem 1:
          Show that the straight lines 2x + y − 9 = 0 and 2x + y − 10 = 0 are parallel.
Solution:
           Slope of the straight line 2x + y − 9 = 0 is m₁ = − 2
           Slope of the straight line 2x + y − 10 = 0 is m₂ = − 2 ∴ m₁ = m₂
           The given straight lines are parallel.

Example problem 2:
      Find the co-ordinates of orthocentre of the triangle formed by the straight lines x − y − 5 = 0, 2x − y − 8 = 0 and 3x − y − 9 = 0
Solution:
      Let the equations of sides AB, BC and CA of a ΔABC be represented by
                x − y − 5 = 0 … (1)
               2x − y − 8 = 0 … (2)
               3x − y − 9 = 0 … (3)
      Solving (1) and (3), we get A as (2, − 3)
       The equation of the straight line BC is 2x − y − 8 = 0. The straight line perpendicular to it is of the form
             x + 2y + k = 0
       A(2, − 3) satisfies the equation (4) ∴ 2 − 6 + k = 0 ⇒ k = 4
       The equation of AD is x + 2y = − 4 … (5)
       Solving the equations (1) and (2), we get B as (3, − 2)
       The straight line perpendicular to 3x − y − 9 = 0 is of the form x + 3y + k = 0
       But B(3, − 2) lies on this straight line ∴ 3 − 6 + k = 0 ⇒ k = 3
       The equation of BE is x + 3y = − 3 … (6)
       Solving (5) and (6), we get the orthocentre O as (− 6, 1).

Example problem 3:
         Find the values of a and b if the equation (a − 4)x² + by² + (b − 3)xy + 4x + 4y − 1 = 0 represents a circle.
Solution:
   The given equation is (a − 4)x² + by² + (b − 3)xy + 4x + 4y − 1 = 0
            (i) coefficient of xy = 0 ⇒ b − 3 = 0 ∴ b = 3
            (ii) coefficient of x² = co-efficient of y² ⇒ a − 4 = b
                        a = 7
                Thus a = 7, b = 3

Practice problems for descriptive geometry:

Practice problem 1:
          If the equation 12x² − 10xy + 2y² + 14x − 5y + c = 0 represents a pair of straight lines, find the value of c. Find the separate equations of the straight lines and also the angle between them.
     Answer: C = 2 ; 6x − 2y + 1 = 0 and 2x − y + 2 = 0 ; tan⁻¹ (1/7)
Practice problem 2:
          Find the equation of the straight line which passes through the given intersection of the straight lines 2x + y = 8 and 3x − 2y + 7 = 0 and is parallel to the straight line 4x + y − 11 = 0
      Answer: 28x + 7y − 74 = 0