Defence Exam Notes: Basic Co-ordinate Geometry

Each of these topics has various sub-topics which require our attention. In this article, you may find a detailed list of the topics with all the necessary formulae which one must keep in mind while preparing these topics. 

Starting with 2D Geometry, the areas of focus are 

• Co-ordinate axes, Cartesian coordinates of points.
• Finding distance between two points A(x1,y1) and B(x2,y2)

• Section formula implementation where a point P divides the line AB in the ratio m:n

• Finding the equation of a line using different methods namely,

(i) Slope form: ‘m’ denotes the slope of the line passing through point A(x1,y1)

(ii) Point form: Line formed by point A(x1,y1) and B(x2,y2)

(iii) Slope intercept form: ‘m’ denotes the slope of the line making ‘c’ intercept on the y-axis.

(iv) Intercept form: ‘a’ intercept and ‘b’ intercept made by the line on x-axis and y-axis respectively.

(v) Normal form: Line is at ‘p’ distance from origin and makes an angle Ɵ. 

• Distance between two parallel lines ax+by+c1=0 and ax+by+c2=0 using the formula

• Distance of line ax+by+c=0 from a point A(x1,y1) using the formula

• Angle between two lines with slope m1 and m2 can be determined using the formula

• Concurrency of lines a₁ x + b₁ y + c₁ = 0 , a₂ x + b₂ y + c₂  = 0 and a₃ x + b₃ y + c₃ = 0

• Collinearity of points A(x1,y1) , B(x2,y2) and C(x3,y3) can be determined using

The general equation for all kinds of Conics structures including circle, ellipse, parabola, hyperbola, and pair of straight lines is ax²+2hxy+by²+2gx+2fy+c = 0.

First, we’ll start with Circle. The key topics here are:

• The eccentricity ‘e’ for circle tends to 0 i.e. e→ 0.
• Equation of circle is given as , where (h, k) is the centre and r is radius.
• For circle, general equation is x²+y²+2gx+2fy+c = 0
• Parametric form of the equation  where r is radius and Ɵ is angle subtended by point on the centre of circle.
• Tangents to the circle which can be determined using the given formula point form, slope form or parametric form.
• Concentric circle problems
• Determining the position of a point with respect to circle
• Problems based on a family of circles, tangents, and intersection of the line with a circle. 

For Ellipse, the following topics must be covered thoroughly:

• Eccentricity ranges between 0 and 1 i.e. 0<e<1
• The general equation of ellipse is  and  
• Determination of major and minor axes, eccentricity, foci and Latus Rectum.
• Problems based on the determination of equation of Directrix.

 For Parabola, one must cover the following points and topics:

• Eccentricity is one i.e. e=1
• Four equations depending on the coordinates of the focus: with focus at  (a,0);  with focus at (-a, 0);  with focus at (0,a) and  with focus at (0,-a)
• Determination of coordinates of Latus Rectum, the equation of Directrix and tangents to the parabola.
• Problems involving intersection of line and parabola
• Problems based on reflection of the parabola 

For Hyperbola, cover the given points:

• Eccentricity is greater than one i.e. e > 1
• The three equations of hyperbola are   where the vertices are  , foci are   and eccentricity is given by   ;    where the vertices are  , foci are  and eccentricity is given by    and   where the vertices are  , foci are   and eccentricity is given by 
• Determination of transverse and conjugate axes.
• Problems based on determination of Directrix
• Problems based on asymptotes, calculation of focal distance and co-normal points
• Problems based on conjugate hyperbola 

When focusing on Pair of straight lines, you must cover problems based on the given points:

• Homogenous equations
• General equations of second degree
• Intercepts of line on axes
• Line theorems and concurrent lines 

All the main key points on which a student must focus are mentioned above. Hence you can prepare them by solving a couple of problems based on each of the above-discussed points. This way you may cover all the important topics in a systematic and easy way which may enhance your chances of solving such questions in the paper correctly

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