Though we are studying physics, it is really necessary to learn mathematical concepts as well because mathematical concepts will help us to solve numerical problems in physics. Some really important mathematical tools are from the topic of coordinate geometry, relation and function, and calculus (differential and integral).
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In this article, we shall discuss the concepts of coordinate geometry and related important questions.
Table of Contents:
Why Maths in Physics?
Physics is all about observing our surroundings and explaining phenomena with the help of laws and theories. Mathematics is an integral part of this, as whenever we need to study processes we need to measure and quantify them. For example, consider the speed of a moving object, density of a substance or force applied to something. To completely study these quantities, we need some number and hence we need maths.
Maths in physics is needed for:
- Quantifying physical quantities.
- Measurement and comparison of physical quantities.
Let us learn about the mathematical tools which are essential to study physics.
Coordinate Geometry
Coordinate geometry is a field of analytic geometry in maths in which analysis is done on the basis of their position in the cartesian plane, also known as a coordinate plane. The position of any point is expressed as coordinates. We shall discuss in detail the concepts related to coordinate geometry.
Coordinate Plane and Axes
Coordinate plane is a two-dimensional plane which has two lines intersecting perpendicularly called the axes. The horizontal line is called the x-axis, whereas the vertical one is called the y-axis. The point of intersection of these axes is called the reference point or origin.The position of any point in this plane is given by taking reference to these two points and origin. The concept of such a plane was first given by the French philosopher and mathematician René Descartes. That is why the coordinate plane is also referred to as the Cartesian plane.
To represent a point A(x, y) on a coordinate plane, we need two coordinates.
- x-coordinate – The perpendicular distance from the y-axis, also known as abscissa.
- y-coordinate – The perpendicular distance from the x-axis, also known as ordinate.
These axes divide the plane into four quadrants.
|
Quadrants of Coordinate Plane |
Sign of the Coordinates |
|---|---|
|
Quadrant I |
(+, +) |
|
Quadrant II |
(–, +) |
|
Quadrant III |
(–, –) |
|
Quadrant IV |
(+, –) |
Shortest Distance Between Two Points
Let A(x1, y1) and B(x2, y2) be two points on the coordinate plane. Then the shortest distance between them is given by the formula:
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\(\begin{array}{l}AB=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\end{array} \)
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This formula is also known as the distance formula.
Slope of a Straight Line
Any straight line drawn on a coordinate plane makes an angle with the horizontal axis, that is the x-axis. The tangent of this angle is called the slope of the straight line. Thus, the slope of a straight line means the steepness of that straight line. Let there be a line segment whose endpoints have the coordinates (x1, y1) and (x2, y2) and make an angle π with the x-axis.
Slope of line = tan π = Perpendicular/ Base = Δy/Δx = (y2 – y1)/(x2 – x1).
Note: (i) The angle π must be measured in anti-clockwise direction.
(ii) If the slope is positive then the straight line is making an acute angle with the x-axis.
(iii) If the slope is negative then the straight line is making an obtuse angle with the x-axis.
(iv) If the slope is zero then the straight line is parallel to the x-axis (as tan 0o = 0).
(v) If the slope is not defined then the straight line is perpendicular to the x-axis (as tan 90o = ∞).
Equation of Straight Line
The equation of a straight is an equation which satisfies each point lying on the given straight line. Thus, to represent any straight line on a coordinate we have the equation
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y = mx + c |
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Where m is the slope of the line and c is the intercept on the y-axis. This equation is also known as the slope-intercept form equation of a straight line.
Note: The equation of a straight line is a linear equation in two variables.
If the line is passing through a point (x1, y1) with a slope m, then the equation of a line is given as
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(y – y1) = m(x – x1) |
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This form of equation of a straight line is known as slope and point form.
If the line is passing through two points (x1, y1) and (x2, y2), then the equation of line is given as
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This form of equation of a straight is known as the two-point form.
If the given straight line cuts intercept a and b on the x and y axes, respectively. Then the equation of the straight line is given as:
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This form of equation of a straight is known as the intercept form.
These are the four forms of equation of a straight line on a coordinate plane.
Special Cases for Slope of a Straight Line
Let there be two straight lines on a coordinate plane with slopes m1 and m2, then we consider the following cases:
|
Type of Straight Lines |
Relation between the Slopes |
|---|---|
|
If the lines are parallel to each other |
m1 = m2 (slopes are equal) |
|
If the lines are perpendicular to each other |
m1 × m2 = –1 |
Equation of a Circle
A circle is the locus of all the points on a coordinate plane which are equidistant to a particular fixed point on the plane. This fixed point is called the centre of the circle and the constant distance from the centre is called the radius of the circle.
The equation of a circle in a coordinate plane is a second degree equation in two variables.
Equation of a circle whose centre is at point (x1, y1) is given by –
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(x – x1)2 + (y – y1)2 = r2, where r is the radius of the circle. |
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Equation of a circle whose centre is at origin is given by –
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x2 + y2 = r2, where r is the radius of the circle. |
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Any point lying on the given circle will have to satisfy the respective equation of the circle.
Recommended Video
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