Intercept

Introduction

• The coordinate graph is also called the Coordinate grid or plane.

• In a coordinate grid, the two perpendicular lines are called the axes.

• The horizontal axis is called the $\mathrm{x\:-\:axis}$ and the vertical axis is called the $\mathrm{y\:-\:axis}$

• In a grid, points are distributed on the number lines, namely, on the $\mathrm{x\:-\:axis}$ and the $\mathrm{y\:-\:axis}$

• The points of contact are written in the ordered pair. By reading the latitude and longitude of the coordinate plane, the location of the points on the grid can be found.

• The points on the $\mathrm{x\:-\:axis}$ is called the $\mathrm{x\:-\:coordinate}$ and the points on the $\mathrm{y\:-\:axis}$ is called the $\mathrm{y\:-\:coordinate}$.

Intercept

• The intercepts are basically the points of intersection in a graph

• If a straight line crosses the x-axis at a point, then that point is called $\mathrm{x\:-\:intercept}$, and the $\mathrm{y\:-\:intercept}$ is the point where a straight line crosses the y-axis

Intercept of a line on axes

When a straight line cuts the $\mathrm{x\:-\:axis}$ at B and the $\mathrm{y\:-\:axis}$ at C, then 𝑂𝐵 and 𝑂𝐶 are known as the intercepts of the line on $\mathrm{x\:-\:axis}$ and $\mathrm{y\:-\:axis}$ respectively

Intercept of a line on conic sections

A straight line intersecting a vertical line at a fixed point rotates about the point forming a surface called a double right circular cone. When the double right circular cone is intersected with a plane two-dimensional curve is formed. Those curves are called conic sections.

There are four conic sections formed by intersection of a plane with a double circular cone.

When the straight line and the conic section is intersected there will be a solution. There will be intercepts represented as ordered pairs.

In a graph, if a straight line is not intersected with the conic section, then there will be no solution. Meaning, no intercepts will be found in the plane.

For example,

A straight line $\mathrm{Ax\:+\:By\:=\:C}$ crosses the hyperbola $\mathrm{\frac{(x\:-\:h)^{2}}{a^{2}}\:-\:\frac{(y\:-\:k)^{2}}{b^{2}}\:=\:1}$

then the straight line will intersect at point $\mathrm{p(x_{1},y_{1})\:and\:Q(x_{2},y_{2})}$ hyperbola, where P and Q are called the points of intersection.

Equation of a line: Slope-Intercept form

When a line crosses the axes, it intersects one or two points in the coordinate graph.

In the above figure, the y-intercept of the given straight line is the distance a.

The distance of the unit a from the origin of the graph is where the straight line cuts the y-axis.

The general equation of line in point form is $\mathrm{y\:-\:y_{1}\:\:=\:m(x\:-\:x_{1})}$

The equation of the straight line with slope m crossing the point (0, 𝑎) is given by

$\mathrm{y\:-\:a\:=\:m(x\:-\:0\:Where\:(0,a)\:is\:(x_{1},y_{1}))}$

$\mathrm{y\:=\:mx\:+\:a}$

Therefore, $\mathrm{y\:=\:mx\:+\:a}$ is the equation of a line in slope-intercept form.

Equation of a line: Intercept form

When a line crosses the axes, it intersects one point in the 𝑥 − 𝑎𝑥𝑖𝑠 and another point in the 𝑦 − 𝑎𝑥𝑖𝑠 of the coordinate graph.

In the above figure, the straight line intersects at two points $\mathrm{(a,0)\:and\:(0,b)}$

The general equation of line in two-point form is $\mathrm{(y\:-\:y_{1})\:=\:\frac{y_{2}\:-\:y_{1}}{x_{2}\:-\:x_{1}}\:(x\:-\:x_{1})}$

Therefore, the equation of the line that intersects at two points $\mathrm{(a,0)\:(x_{1},y_{1})\:and\:(0,b)\:=\:(x_{2},y_{2})}$

$\mathrm{is\:(y\:-\:0)\:=\:\frac{b\:-\:0}{0\:-\:a}\:(x\:-\:a)}$

$\mathrm{-\:ay\:=\:bx\:-\:ab}$

$\mathrm{\Rightarrow\:bx\:+\:ay\:=\:ab\:.........(1)}$

Dividing (1) by 𝑎𝑏, we get

$\mathrm{\frac{bx}{ab}\:+\:\frac{ay}{ab}\:=\:\frac{ab}{ab}}$

By cancelling the common terms, we obtain the equation of line in intercept form

$\mathrm{\frac{x}{a}\:+\:\frac{y}{b}\:=\:1}$

Solved Examples

1. From the figure given below, find the intercepts of the line.

Solution

The line cuts the x-axis a $\mathrm{(4,0)}$ and the y-axis at $\mathrm{(0,3)}$

Therefore, the intercepts of the line on the axes are $\mathrm{(4,0)\:and\:(0,3)}$

2. In the given straight line $\mathrm{7x\:+\:9y\:=\:14}$, find the x-intercept and the yintercept.

Solution −

The equation of the straight line is given by $\mathrm{7x\:+\:9y\:=\:14}$

To find the x-intercept of the given line substitut $$

$$\mathrm{7x\:+\:9(0)\:=\:14}$$

$$\mathrm{\Rightarrow\:7x\:=\:14}$$

$$\mathrm{x\:=\:2}$$

To find the y-intercept of the given line substitute $\mathrm{x\:=\:0}$

$$\mathrm{7(0)\:+\:9y\:=\:14}$$

$$\mathrm{\Rightarrow\:9y\:=\:14}$$

$$\mathrm{y\:=\:\frac{14}{9}}$$

Therefore, the x-intercept = 2 and y-intercept $\mathrm{=\:\frac{14}{9}}$

If the slope of the straight line passing through $\mathrm{(1,2)\:is\:-\frac{5}{7}}$

3.Find the equation of the line

Solution −

The equation of the line in slope-intercept form is $\mathrm{y\:=\:mx\:+\:a}$

Substituting $\mathrm{m\:=\:-\:\frac{5}{7}\:in\:y\:=\:mx\:+\:a}$ we obtain

$\mathrm{2\:=\:-\:\frac{5}{7}\:(1)\:+\:a}$

$\mathrm{14\:=\:-5\:+\:a}$

$\mathrm{\Rightarrow\:a\:=\:19}$

Therefore, the equation of the line is $\mathrm{7y\:=\:-5x\:+\:19}$

4.If the straight line is $\mathrm{6x\:+\:3y\:=\:12}$. Represent the equation of the line in intercept form and find the $\mathrm{x\:-\:intercept}$ and the $\mathrm{y\:-\:intercept}$.

Solution −

The equation of the line is given by $\mathrm{6x\:+\:3y\:=\:12.......(1)}$

By dividing (1) by 12 on both the sides, we get

$\mathrm{\frac{6x}{12}\:+\:\frac{3y}{12}\:=\:\frac{12}{12}}$

$\mathrm{\frac{x}{2}\:+\frac{y}{4}\:=\:1}$ is the equation of the line in intercept form

Therefore, the $\mathrm{x\:-\:intercept\:=\:2\:and\:the\:y\:-\:intercept\:=\:4}$

Conclusion

• In a coordinate graph, the horizontal axis is called the $\mathrm{x\:-\:axis}$ and the vertical axis is called the $\mathrm{y\:-\:axis}$

• When a straight line crosses the x-axis at a point then that point is called $\mathrm{x\:-\:intercept}$, and the $\mathrm{y\:-\:intercept}$ is the point where a straight line crosses the y-axis

• The points on the $\mathrm{x\:-\:axis}$ is called the $\mathrm{x\:-\:coordinate}$ and the points on the $\mathrm{y\:-\:axis}$ is called the $\mathrm{y\:-\:coordinate}$

• The equation of a straight line is $\mathrm{\frac{x}{a}\:+\:\frac{y}{b}\:=\:1}$ where a and b are the $\mathrm{x\:-\:intercept}$, and the $\mathrm{y\:-\:intercept}$ of the line

• The equation of a line given by slope-intercept form is $\mathrm{y\:=\:mx\:+\:a}$

FAQs

1. Who introduced the coordinate system?

• The coordinate system, also called the Cartesian Plane, is named after the French Mathematician, René Descartes.

• He was the one who introduced the solving method of geometric problems in a coordinate plane.

• He is known as the father of coordinate geometry.

2. Who invented conic sections?

The Greek Mathematician Menaechmus introduced the concept of conic sections

3. What is the general equation of the circle and how is it formed from the double circular cone?

The general equation of the circle is $\mathrm{(x\:-\:h)^{2}\:-\:(y\:-\:k)^{2}\:=\:a^{2}}$

where a is the radius and (h, k) is the centre of the circle.

• The bottom surface of the conic section is a circle

• When the plane is intersected horizontally with the double circular cone, a circle is formed

• And when the plane is perpendicular to the axis the ellipse is changed into a circle

4. What are called degenerate conics?

If a plane intersects at the vertex of the double circular cone, then the formed figures are called degenerate conics

5. Give some real life applications of coordinate geometry

• To ensure flight safety, coordinate geometry is used to update location of flights

• It is used to gather information on the position of satellites.

• The latitude and longitude of the earth can be described by the coordinate system.

• The prediction of the future paths of a storm can be described with the help of coordinate geometry.

• The location of cities and states in a map are positioned with the help of coordinate geometry