Coordinate Geometry Formulas, Solutions, Study Material

Coordinate Geometry important questions on Page No - 6

We studied about coordinate axes, coordinate plane, plotting of points in a plane, distance between two points, section formula, etc. All these concepts are the basics of coordinate geometry. Let us have a brief recall of coordinate geometry done in earlier classes. To recapitulate, the location of the points (6, – 4) and (3, 0) in the X-plane is shown in (Fig 10.1).

We may note that the point (6, – 4) is at 6 units distance from the y -axis measured along the positive x -axis and at 4 units distance from the X –axis measured along the negative Y-axis. Similarly, the point (3, 0) is at 3 units distance from the Y –axis measured along the positive X -axis and has zero distance from the X -axis. We also studied.

1. Distance between the points $P(x_{1}y_{1})\: and\: Q(x_{2}y_{2})$

$PQ\: =\: \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

2. The coordinates of a point dividing the line segment joining the points $(x_{1}, y_{1}\:and\: x_{2}, y_{2}$ internally, in the ratio m: n are $\frac{mx_{2}+nx_{1}}{m+n}\: ,\: \frac{my_{2}+my_{1}}{m+n}$

Example: he coordinates of the point which divides the line segment joining A (1, –3) and B (–3, 9) internally, in the ratio 1: 3 are given by $x\: =\: \frac{1(-3)+3.1}{1+3}\:=\:0$ and

3. In particular, if m=n ,the coordinate of the mid point of the line segment joining the point $(x_{1}, y_{1}\:and\: x_{2}, y_{2}$ . Are $\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}$

4. Area of the triangle whose vertices are $(x_{1},y_{1}),(x_{2},y_{2}),(x_{3}y_{3})\: is\: \frac{1}{2}\left | x_{1} (y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})\right|$
Note : If the area of the triangle ABC is zero , then three point A,B and C lie on a line , i.e ,they are collinear .
We shall continue the study of coordinate geometry to study properties of the simplest geometry figure –straight line. Despite its simplicity , the line is a virtual concept of geometry. and enters into our daily experience in numerous interesting and useful ways. Main focus is on representing ,the slope is most essential.
Slope of a Line: A line in a coordinate plane forms two angles with the x-axis ,which are supplementary The angle (say)
Θ made by the line l with positive direction of x -axis and measured anti clockwise is called the inclination of the line. Obviously 0° ≤ θ ≤ 180° (Fig 10.2) .
We observe that lines parallel to x -axis, or coinciding with x -axis, have inclination of 0°. The inclination of a vertical line (parallel to or coinciding with y -axis) is 90°

Definition: If θ is the inclination of a line l , then tan θ is called the slope or gradient of the line l . The slope of a line those inclination is 90° is not defined. The slope of a line is denoted by m . Thus, m = tan θ, θ ≠ 90° It may be observed that the slope of x -axis is zero and slope of y -axis is not defined.