Analytical Geometry Class 11 Mathematics Solutions | Exercise - 9.2

  Chapter - 9  
Analytical Geometry

We all have to study Analytical Geometry chapter in class 11 mathematics. Though we study about this chapter, many of us don't know what exactly 'Analytical Geometry means. So let me tell you about this chapter a little bit. 

Basically Analytical Geometry is a branch of algebra which deals with the modelling of some geometrical objects such as curves, lines,plots and so on. In this chapter we will study about various subtopics falling under it. So let's have a quick look at them too.

Exercise - 13.1

Review of Equations of Straight Lines 

The equation of the line parallel to x-axis is x = a When a = 0, x = 0 will be the equation of y-axis. In the same way, the equation of the line parallel to y-axis is y = b. If b = 0 , then y = 0 will be the equation of the x-axis.

Three standard forms of equation of a line

Three standard forms of equation of a line are given below:

• y = mx + c, is known as the slope-intercept form where m and c are the slope and the y- intercept respectively.
• x/a + y/b = 1, is known as the double-intercept form where a and b are the x-intercept and y- b intercept respectively.
• xcosα + ysinα = p is known as the normal form or perpendicular form where p is the perpendicular from origin to the line and a, the angle made by the perpendicular with x- axis.

Equations of lines in spacial cases:

The following are the two special forms of a line:

• Point slope form: y - y₁ = m(x - x₁) where m is the slope of the line through (x₁, y₁).
• Two points form: y - y₁ = {(y₂ - y₁) / (x₂ - x₁)} (x - x₁) is the line through two given points (x₁, y₁) and (x₂,y₂)

Linear Equation

The general equation of first degree in x and y is Ax + By + C = 0 where A, B and C constants and A, B are not simultaneously zero. This equation is called the linear equation. This equation always represents a straight line

Important Points of Concurrencies

The following are the four important definitions of point of concurrencies of three straight lines.

i) Orthocentre: The perpendiculars drawn from the vertices on their opposite sides meet at a point. This point is known as 'Orthocentre'.

ii) Circumcentre: The perpendicular bisectors of the sides of a triangle meet at a point. This point is known as 'circumcentre'.

iii) Incentre: The bisectors of the internal angles of a triangle meet at a point. This point is known as 'Incentre'. iv) Centroid: The medians of a triangle meet at a point. This point is known as 'centroid'.

The Two Sides of a Line

Let P (x,y,) and Q(x, y) be any two points and Ax+By+C=0 a line.

Join PQ. Let PQ (produced if necessary) meet the given line at R. Let PR:RQ=mn, m:n is positive if R divides PQ internally i.e. P and Q are on opposite sides of the line (Fig (a)); and m:n is negative if R divides PQ externally, i.e. P and Q are on the same sides of the line (Fig. (b)).