Kerala State Board New Syllabus Plus One Maths Notes Chapter 10 Straight Lines.
Kerala Plus One Maths Notes Chapter 10 Straight Lines
I. Slope of Line
The slope of a line is the ‘tan’ of the angle the line makes with the positive direction of the x-axis. If θ is the angle then, slope = tan θ.
The slope of the x-axis is zero and that of the y-axis is not defined.
Parallel lines have the same slope.
The product of the slopes of perpendicular lines is -1.
The slope is positive if θ < 90°. The slope is negative if θ > 90°.
The slope of a line passing through two points (x1, y1) and (x2, y2) is \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
If three points A, B, and C are collinear, then AB and BC have the same slope.
If m1 and m2 be slopes of two lines then, θ the angle between is given by tan θ = \(\left|\frac{m_{2}-m_{1}}{1+m_{1} m_{2}}\right|\), 1 + m1m2 ≠ 0
II. Equation of a Line
Equation of x-axis is y = 0.
Equation of y-axis is x = 0.
The equation of a horizontal line is y = a. If ‘a’ is positive then the line is above the x-axis and if negative it will be below the x-axis.
The equation of a vertical line is x = a. If ‘a’ is positive then the line is to the right of the x-axis and if negative it will be to the left of the x-axis.
Point-slope form: y – y1 = m(x – x1), where ‘m’ is the slope and (x1, y1) is a point on the line.
Two-Point form:
y – y1 = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) (x – x1) where (x1, y1) and(x2, y2) are two point on the line.
Slope intercept form:
1. y = mx + c, where ‘m’ is the slope and ‘c’ is the y-intercept.
2. y = m(x – d), where ‘m’ is the slope and ‘d’ is the x-intercept.
Intercept form: \(\frac{x}{a}+\frac{y}{b}=1\) = 1, where ‘a’ and ‘b‘ are x and y intercept respectively.
Normal form: x cos θ + y sin θ = p, where ‘p’ is the length of the normal from the origin to the line and ‘θ’ is the angle the normal makes with the positive direction of the x-axis.
General equation of a Line: ax + by + c = 0, where a, b and c are real constants.
1. Slope of the line ax + by + c = 0 is \(-\frac{a}{b}\)
2. Parallel lines differ in constant term, i.e; a line parallel to ax + by + c = 0 is ax + by + k = 0.
3. A line perpendicular to ax + by + c = 0 is bx – ay + k = 0.
4. The equation of the family of lines passing through the intersection of the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is of the form a1x + b1y + c1 + k(a2x + b2y + c2) = 0.
5. The perpendicular distance of a point (x1, y1) from the line ax + by + c = 0 is \(\left|\frac{a x_{1}+b y_{1}+c}{\sqrt{a^{2}+b^{2}}}\right|\)
6. The distance between the parallel lines ax + by + c = 0 and ax + by + k = 0 is \(\left|\frac{c-k}{\sqrt{a^{2}+b^{2}}}\right|\)
7. Normal form of the equation ax + by + c = 0 is x cos θ + y sin θ = p;
Where cos θ = \(\pm \frac{a}{\sqrt{a^{2}+b^{2}}}\); sin θ = \(\pm \frac{b}{\sqrt{a^{2}+b^{2}}}\) and p = \(\pm \frac{c}{\sqrt{a^{2}+b^{2}}}\)
Proper choice of signs is made so that p should be positive.
III. Shifting of Origin
An equation corresponding to a set of points with reference to a system of coordinate axes by shifting the origin is shifted to a new point is called a translation of axes.
Let us take a point P (x, y) referred to the axes OX and OY. Let (h, k) be the coordinates of origin and P(X, Y) be the coordinate of P(x, y) with respect to the new axis. Then, the transformation relation between the old coordinates (x, y) and the new coordinates (X, Y) are given by X = x + h and Y = y + k.