1. DISTANCE BETWEEN TWO POINTS : If there are two points \(A (X_1, Y_1)\) and \(B (X_2, Y_2)\) on the XY plane, the distance between them is given by \(AB = d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Distance of the point \(P(x, y)\) from the origin O is \(OP = \sqrt{(x^2+ y^2)}\)

2. The coordinates of \(P(X,Y)\) such that P divides the line joining A (X, Y1) and B (X2, Y2) internally I the ratio m : n will be : \(X=\frac{mX_2+nY_1}{m+n}, Y=\frac{mY_2+nY_1}{m+n}
\)

3. If \(P (X,Y)\) is the midpoint between A & B then \(X = \frac{(X_1+ X_2) }{2} , Y = \frac{(Y_1+ Y_2) }{2}\)


EQUATION OF A CURVE :

An equation in two variables X and Y (with or without a constant term) is called the equation of a particular curve if the graph of that equation plotted on the XY cartesian plane gives that particular curve. e.g. \(X^2 + Y^2 = 36\). If we plot this curve, by taking different values of Y (and thereby different values of X), we get a circle. Hence \(X^2 + Y^2 = 36\) is the equation of a circle.

STRAIGHT LINE :

An equation of the form AX + BY + C = 0 is called the general equation of a straight line, where X and Y are variables and A, B, C are constants. Any point lying on this line will satisfy the equation of the line. i.e. the coordinates of the point when substituted by X & Y resp. in the above equation will make the LHS vanish.


GENERAL CONCEPTS

If AB is a straight line on the XY plane, then the ratio of y-intercept to x-intercept (with signs) is called its slope and is denoted by ‘m’. The lengths OP and OQ are respectively called the intercepts on X and Y axes, made by the line.




The same straight line can be represented by many different forms of the same equation.

1. If ‘m’ is the slope of the line and ‘c’ the intercept made by the line on Y axis, the equation is \(Y = mX + c\)

2. If the slope of the line is m and it passes through \((X_1, Y_1)\), the equation is \((Y – Y_1) = m (X – X_1)\)

3. If the line passes through two points \((X_1, Y_1)\) and \((X_2, Y_2)\), the equation is

\(Y-Y_1=\frac{Y_2-Y_1}{X_2-X_1} (X-X_1)\)

Hence the slope of the line through \((X_1, Y_1), (X_2, Y_2)\) is given by \(m = \frac{(Y_2 – Y_1) }{(X_2 – X_1)}\) or \(\frac{(Y_1 – Y_2)}{(X_1 – X_2)}\)


4. If the X intercept of the line is a and Y intercept is b, the equation is : \(\frac{ X}{a }+ \frac{Y}{b} = 1
\)

5. General form : \(A_X + B_Y + C = 0\)

In this slope \(= m = –\frac{A}{B}\)

X intercept \(= –\frac{C}{A }\)

Y intecept \(= –\frac{C}{B}\)

(a) If two lines are parallel then their slopes are equal \((m_1= m_2)\). If two lines do not intersect, they are parallel.
(b) If two lines are perpendicular to each other, the product of their slopes is –1. \((m_1 m_2 = –1)\).
(c) \(a_1X + b_1Y + c_1 = 0 \) and \(a_2X + b_2Y + c_2 = 0\) will represent the same straight line if \(\frac{a_1}{a_2} = \frac{b_1}{b_2 }= \frac{c_1}{c_2}\) . In this case, the lines are coincident and theoretically intersect at infinite points.

6. The point of intersection of two lines (X,Y) is obtained by simultaneously solving both the equations.

7. The equation of a line parallel to a given line \(AX + BY + C = 0\), will be \(AX + BY + K = 0\) where K is a constant which can be found by any additional given condition.

8. The equation of a line perpendicular to a given line \(AX + BY + C = 0\) will be \(BX – AY + K = 0\), where K is a constant which can be found by additional given conditions.

9. The length of perpendicular (p) from \((X_1, Y_1)\) on the line \(AX + BY + C = 0\) is :





10. Equation of a line parallel to X axis is Y = b (b is a constant)

11. Equation of a line parallel to Y axis is X = a (a is a constant)

12. Equation of X and Y axes are Y = 0 and X = 0 respectively

13. Any point on the X-axis can be taken as (a,0)

14. Any point on the Y-axis can be taken as (0,b)

15. In order to find the X–intercept of a line, put Y = 0 in the equation of the line and find X

16. In order to find the Y-intercept of a line, put X = 0 and find Y

17. The image of the point (a, b) in the x-axis is (a, -b)

18. The image of the point (a, b) in the y-axis is (-a, b)

19. The image of the point (a, b) in the line y = x is (b, a)

20. To plot a line, first put y = 0, find the point on the x-axis; then put x = 0, fins the point on the y-axis. Join the two points to get the desired graph.

Note :
The equation of a circle with centre (h, k) and radius r is \((x – h)^2+ (y – k)^2= r^2\).
The equation of a circle with centre (0, 0) and radius r is \(x^2+ y^2= r^2\).
The equation \(y = Ax^2+ Bx + C\) is the equation of the quadratic graph which is a parabola with an axis parallel to the y-axis. If \(A > 0\), the parabola opens upwards. If \(A < 0\), the parabola opens downwards.
If \(B^2> 4AC\), the parabola cuts the x-axis at 2 different points.
If \(B^2= 4AC\), the parabola touches the x-axis at one point (the two points become co-incident).
If \(B^2< 4AC\), the parabola does not cut the x-axis at all.