Straight lines and conic sections notes

STRAIGHT LINES
CONCEPTS AND RESULTS
** Any point on the X-axis is (x, 0) and on the Y-axis is (0, y)
** Distance between two points A(x1 , y1) & B(x2 , y2) is AB = (x 2  x1 )2  (y2  y1 )2 .
** Section formula
(i) Coordinates of a point dividing the line joining A(x1 , y1) & B(x2 , y2) internally in the ratio m
: n is
 mx 2  nx1 my2  ny1  .
 , 
 mn mn 
(ii) Coordinates of a point dividing the line joining A(x1 , y1) & B(x2 , y2) externally in the ratio m :
n is
 mx 2  nx1 my2  ny1  .
 , 
 mn mn 
 x  y1 x 2  y 2 
** Coordinates of the mid point of the line joining A(x1 , y1) & B(x2 , y2) is  1 , 
 2 2 
** Centroid of a ABC with vertices A(x1 , y1), B(x2 , y2) & C(x3 , y3)  x1  x 2  x 3 , y1  y2  y3  .
 3 3 
** In centre of ABC with vertices A(x1 , y1), B(x2 , y2) & C(x3 , y3) is
 ax1  bx 2  cx3 ay1  by 2  cy3 
 ,  where a = BC, b = AC, c = AB.
 abc abc 
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** Area of ABC with vertices A(x1, y1), B(x2, y2) & C(x3, y3) = | x1(y2 – y3) + x2(y3 – y1) + x3(y1 –
2
y2)|.
** Equation of any line parallel to X-axis is y = a, & equation of X-axis is y = 0.
** Equation of any line parallel to Y-axis is x = b & equation of Y axis is x = 0.
** Slope of line inclined at an angle  with the + ve X- axis = tan .
** Slope of a line parallel to X-axis = 0 , slope of a line parallel to Y-axis = undefined.
Slope of a line equally inclined to the coordinate axes is –1 or 1.
y 2  y1
** Slope of a line joining the points A(x1, y1), B(x2, y2) is , x1  x2.
x 2  x1
a
** Slope of the line ax + by + c = 0, is  .
b
** If two lines are parallel, then their slopes are equal.
** If two lines are perpendicular, then the product of their slopes is –1.
** Any equation of the form Ax + By + C = 0, with A and B are not zero, simultaneously, is called the
general linear equation or general equation of a line.
(i) If A = 0, the line is parallel to the x-axis (ii) If B = 0, the line is parallel to the y-axis
(iii) If C = 0, the line passes through origin.
** Equation of a line having slope = m and cutting off an intercept „c‟ and Y-axis is y = mx + c.
** Equation of a line through the point (x1, y1) and having slope m is y – y1 = m(x – x1).
x y
** Equation of a line making intercepts of „a‟ & „b‟ on the respective axes is   1
a b
** The equation of the line having normal distance from origin p and angle between normal and the
positive x-axis ω is given by x cosω + y sinω = p .
ax1  by 1  c
** Distance of a point P(x1, y1) from the line ax + by + c = 0 is d = .
a 2  b2

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, ** Equation of the line parallel to ax + by + c = 0 is ax + by + λ = 0.
** Equation of the line perpendicular to ax + by +c= 0 is bx – ay + λ = 0.
m 2  m1
** If two lines are intersecting and  is the angle between them, then tan  = where m1 =
1  m1m 2
slope of first line, m2 = slope of second line and  = acute angle.
If tan  = negative   = obtuse angle between the intersecting lines.
c1  c 2
** Distance between two parallel lines ax + by + c1 = 0 & ax + by + c2 = 0 is .
a 2  b2



CONIC SECTIONS

CONCEPTS AND RESULTS
CIRCLES

A circle is a locus of a point which moves in a plane such that its distance from a fixed point in that
plane in always constant. The fixed point is said to be the circle and the constant distance is said to be
the radius.
** The equation of a circle with centre (h , k) and the radius r is (x – h)2 + (y – k)2 = r2.
** The equation of a circle with centre (0 , 0) and the radius r is x2 + y2 = r2.
** General equation of a circle is x2 + y2 + 2gx + 2fy + c = 0 whose centre is (–g, –f) and radius is
g2  f 2  c
** Equation of a circle when end points of diameter as A(x1, y1), B(x2, y2) is given by
(x – x1) (x – x2) + (y – y1) (y – y2) = 0.
** Length of intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 on the X and Y-axes are
2 g 2  c and 2 f 2  c .
CONICS

Conic Section or a conic is the locus of a point which moves so that its distance from affixed point
bears a constant ratio to its distance from a fixed line.

The fixed point is called the focus, the straight line the directrix and the constant ratio denoted
by e is called the eccentricity.
distance between P(x, y) & Focus
Eccentricity (e) e= .
distance between P(x, y) & Directrix
If e = 1, then conic is a parabola.
If e < 1, then conic is a an ellipse.
If e > 1, then conic is a hyperbola.
If e = 0, then conic is a circle.




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