(x, y), (-x, y), (-x, -y) and (x, -y) will be the co-ordinates of the points in I, II, III and IV quadrants respectively, if x and y are positive numbers.
Let any two points on the plane be A(\(x_{1}, y_{1}\)) and B(\(x_{2}, y_{2}\)).Then the distance between A and B is represented as AB i.e. given by
AB = \(\sqrt{(x_{2} – x_{1})^2 + (y_{2} – y_{1})^2}\)
The distance of the point A(x, y) from the origin (0, 0) is given by
\(\sqrt{x^2 + y^2}\)
Internal division:
The co-ordinates of the point P which divides the line joining AB internally in the ratio m: n are
[\(\frac{mx_{2} + nx_{1}}{m + n}\), \(\frac{my_{2} + ny_{1}}{m + n}\)]
Here, A and B are two points i.e. A(\(x_{1}, y_{1}\)) and B(\(x_{2}, y_{2}\)).
External division:
The co-ordinates of the point P which divides the line joining points A(\(x_{1}, y_{1}\)) and B(\(x_{2}, y_{2}\)) externally in the ratio m: n are
[\(\frac{mx_{2} + nx_{1}}{m + n}\), \(\frac{my_{2} + ny_{1}}{m + n}\)]
Here, the point P is beyond A and B for external division and it can be either close to B or close to A.
Mid point:
If P is the mid point and A(\(x_{1}, y_{1}\)) & B(\(x_{2}, y_{2}\)) are the two points given then m: n = 1: 1 i.e. m = n.
Therefore, the co-ordinates or the point P are given by
(\(\frac{x_{1} + x_{2}}{2}\), \(\frac{y_{1} + y_{2}}{2}\))
Centroid:
If A, B and C are the vertices of a triangle. The three medians of the triangle intersect at its centroid and the centroid divides the median in the ratio 2 : 1.
Co-ordinates of the point G are given by
(\(\frac{x_{1} + x_{2} + x_{3}}{3}\), \(\frac{y_{1} + y_{2} + y_{3}}{3}\))
Straight line:
Eg: 2x + 3y + 7 = 0
Slope of a line:
Parallel lines: If the slopes of two lines are equal, then the two lines are parallel.
Perpendicular lines: If and only if the product of the two lines slopes is -1, then the two lines are perpendicular.
Equation of lines in standard forms:
1. Line’s general form:
2. Line’s slope intercept form: The equation of a straight line having slope ‘m’ and making an intercept ‘c’ with y-axis is
y = mx + c.
3. Line’s point form: The equation of a straight line passing through the points (\(x_{1}, y_{1}\)) and (\(x_{2}, y_{2}\)) is
(\(y – y_{1}\)) = \(\frac{y_{2} – y_{1}}{x_{2} – x_{1}}(x – x_{1}\))
Where, \(x_{1}\) ≠ \(x_{2}\)
4. Line’s point slope form: The equation of a straight line passing through the point (\(x_{1}, y_{1}\)) and having slope m is given by
(\(y – y_{1}\)) = m\((x – x_{1}\))
5. Line’s intercept form: The equation of a line making intercepts a and b on the x and y axis respectively is given by
\(\frac{x}{a} + \frac{y}{b}\) = 1
2. What is the equation of the line passing through the point (2, 3) and perpendicular to the line 5x + 4y + 6 = 0.
Solution:
3. A line is represented by the equation y = 3x + 2.
Quantity A | Quantity B |
The slope of the line | The y-intercept of the line |
A. The quantity on the left is greater.
B. The quantity on the right is greater.
C. Both are equal.
D. The relationship cannot be determined without further information.
Solution:
4. The figure of a rhombus is as shown below:
Quantity A | Quantity B |
The length of line segment EB | The length of line segment AB |
A. The quantity on the left is greater.
B. The quantity on the right is greater.
C. Both are equal.
D. The relationship cannot be determined without further information.
Solution:
5.
Quantity A | Quantity B |
The distance between the points with rectangular coordinates (0,5) and (0,10) | The distance between the points with rectangular coordinates (1,8) and (-3,5) |
A. The quantity on the left is greater.
B. The quantity on the right is greater.
C. Both are equal.
D. The relationship cannot be determined without further information.
Solution: