Profile
Settings
Refer your friends
Sign out
Equation of pair of straight lines
A straight line often called a line, is an infinite one-dimensional shape with no width and only length. A straight line consists of an infinite number of points. Two variables use linear equations to represent a straight line. Under certain conditions, two variables use a quadratic equation to represent a pair of straight lines.
Equations of two or more lines can be represented together by equations of degrees greater than 1. Since we know that the linear equations of x and y represent straight lines, the product of the two linear equations represents two straight lines, that is, a pair of straight lines. Therefore, we examine a pair of straight lines as quadratic equations of x and y.
Suppose, L1 = a1x + b1y + c1 = 0
And, L2 = a2x + b2y + c2 = 0
If, P (x1, y1) is a point on L1, which satisfies the equation L1 = 0
Similarly, P (x1, y1) is on L2 then L2 = 0
Therefore, L1 * L2 = 0
y + √3x = 0
y – √3x = 0
combining the above equation,
(y + √3x) (y – √3x) = 0
- y2 – 3x2 = 0, this represents pair of straight lines
Angle pairs
Angle pairs are the angles that are displayed in pairs to represent a particular geometric property.
Types of angle pairs are:
Complementary angles: If the sum is 90 ° (right angle), the two angles are complementary. This means <A + <B = 90°
Supplementary angles: ∠ABD is a complement to ∠DBC. Complementary angles are pairs of angles whose sum of degrees is equal to 180 ° (straight line).
Vertical angles: The apex angle is the pair of angles formed by two intersecting lines so that the angles are opposite to each other.
Alternate interior angles and alternate exterior angles: Alternating internal angles are the paired angles formed when a line intersects two parallel lines. Alternating internal angles are always equal to each other. The alternate exterior angle is the apex angle of the alternate interior angle. The alternative exterior angles are equivalent.
Corresponding angles: The corresponding angle is the angle of the pair formed when the line intersects the pair of parallel lines. The corresponding angles are also equal to each other.
Adjacent angles: Adjacent angles are the angles of pairs that are adjacent to each other. They share a common vertex and a common edge.
Formulas of pairs of straight lines
The angle between pair of straight lines
Consider the equation of a pair of straight lines passing through the origin:
ax2 + 2hxy + by2 = 0
let m1 and m2 be the slopes of these two lines
by dividing x2 and substituting y/x = m
we get, bm2 + 2hm + a = 0
thus, m1 + m2 = -2h/b and m1m2 = a/b
if the angles between the two lines is Θ then,
tan Θ = ǀ m2 – m1 / 1 + m2m1 ǀ
= ǀ √(m2 – m1)2 – 4m1m2 / 1 + m2m1 ǀ
= ǀ √( -2h/b)2 – 4 a/b / 1 + a/b ǀ
tan Θ = ǀ 2√h2 – ab / a + b ǀ
As a result of this equation, we can conclude that if m1 and m2 are different in real numbers, that is, h2 − ab> 0, the lines match in real numbers. If they are equal in real numbers, that is, if h2 − ab = 0, then If m1 and m2 are not real numbers, that is, if h2 – ab < 0, the line is not a real number (imaginary number).
The general equation in quadratic form
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
two lines can be coincident or parallel only when, tan Θ = 0 i.e., h2 – ab = 0
two lines can be perpendicular only when tan Θ is not defined i.e., a + b = 0
two lines can equally inclined to axes only when, the coefficient of xy = 0 i.e., h=0
Conclusion
A pair of straight lines can also be represented as the product of two linear equations, x and y, that represent the straight lines. The concept of straight-line pairs is very useful in the mathematical world because it simplifies complex problems more easily. In this article, you will learn how to create straight-line pairs, general formulas for straight-line pairs, and related formulas.
