Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 11 Straight Lines will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 11 Straight Lines

Distance formula:

Distance between two points A (x_{1}, y_{1}) and A (x_{2}, y_{2}) = \(\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\)

Section Formula:

If C(x, y) divides the join of A (x_{1}, y_{1}) and A (x_{2}, y_{2}) in the ratio m: n internally then, x = \(\frac{m x_2+n x_1}{m+n}\), y = \(\frac{m y_2+n y_1}{m+n}\)

Note:

- If the division is external then, x = \(\frac{m x_2-n x_1}{m-n}\), y = \(\frac{m y_2-n y_1}{m-n}\)
- If C(x, y) is the midpoint then x = \(\frac{x_1+x_2}{2}\), y = \(\frac{y_1+y_2}{2}\)

Area of triangle formula:

The area of triangle with vertices A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) is given by = \(\frac{1}{2}\)[x_{1}(y_{2} – y_{3}) + x_{2}(y_{3} – y_{1}) + x_{3}(y_{1} – y_{2})]

Different points related to a triangle:

(a) Centroid of the triangle with vertices A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) is = G \(\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)\)

(b) In centre of a triangle with vertices A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) is = I \(\left(\frac{a x_1+b x_2+c x_3}{3}, \frac{a y_1+b y_2+c y_3}{3}\right)\)

**Slope Of A Line:**

(a) Angle of inclination: the angle θ made by a line with positive x-axis is the angle of inclination.

(b) Slope of a line: Slope of a line is the tangent of angle of inclination. i,.e m = tan θ.

(c) Slope of a line joining A(x_{1}, y_{1}), and B(x_{2}, y_{2}) = \(\frac{y_2-y_1}{x_2-x_1}\)

Note:

(i) Slope of x-axis = 0

Slope of any line parallel to x-axis = 0

(ii) Slope of y-axis = ∞

Slope of any line parallel to y-axis = ∞

Angle Between two Lines:

Angle Φ between two lines with slope m_{1} and m_{2} is given by tan Φ = \(\pm \frac{\left(m_1-m_2\right)}{1+m_1 m_2}\)

Note:

- To find the acute angle between two lines use the formula. tan Φ = \(\left|\frac{m_1-m_2}{1+m_1 m_2}\right|\)
- Two lines are parallel if m
_{1}= m_{2} - Two lines are perpendicular if m
_{1}m_{2}= (-1).

**Collinearity Of Three Points:**

Three points A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) are collinear if

(i) Sum of distances between two pairs of points = Distance between the 3rd pair.

Or, (ii) Area of Δ ABC = 0

Or, (iii) Let B(x_{2}, y_{2}) divides the join of AC in ratio k: 1

∴ \(x_2=\frac{k x_3+x_1}{k+1}, y_2=\frac{k y_3+y_1}{k+1}\)

The value of k obtained from two cases are equal.

Or, (iv) Slope of AB = Slope of AC.

Equation of a straight line:

Lines parallel to co-ordinate axes:

(i) Equation of any line parallel to x-axis is, y = k

⇒ Equation of x-axis is, y = 0

(ii) Equation of any line parallel to y-axis is, x = k

⇒ Equation of y-axis is, x = 0

**Lines Not Parallel To Any Axes:**

(i) Slope intercept form:

Equation of a line with slope ‘m’ and y-intercepts ‘c’ is: y = mx + c

(ii) Point slope form:

Equation of a line with slope ‘m’ and passing through a point A(x_{1}, y_{1}) is: y – y_{1} = m(x – x_{1})

(iii) Two point form:

Equation of the line passing through A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is : \(\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}\)

(iv) Intercept form:

Equation of a line with x-intercept ‘a’ and y-intercept ‘b’ is \(\frac{x}{a}+\frac{y}{b}=1\)

(v) Normal form:

Equation of a line whose distance form origin is P and the perpendicular drawn form origin to the line makes an angle α with positive direction of x-axis is: x cos α + y sin α = P

(vi) Parameteric form or symmetric form:

Equation of the line passing through A(x_{1}, y_{1}) and making an angle θ with positive direction of x-axis is: \(\frac{x-x_1}{\cos \theta}=\frac{y-y_1}{\sin \theta}\) = r

Or, x = x_{1} + r cos θ, y = y_{1} + r sin θ

where r = The directed distance between points P(x, y) and A(x_{1}, y_{1})

(vii) General form:

General equation of a straight line is Ax + By + C = 0

Note:

- Slope of this line = –\(\frac{\mathrm{A}}{\mathrm{B}}\)
- x-intercept = –\(\frac{\mathrm{C}}{\mathrm{A}}\)
- y-intercept = – \(\frac{\mathrm{C}}{\mathrm{B}}\)
- Two lines a
_{1}x + b_{1}y + c_{1}= 0 and a_{2}x + b_{2}y + c_{2}= 0 are parallel if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\) perpendicular if a_{1}a_{2}+ b_{1}b_{2}= 0 and coincident if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)

Condition of concurrency of three lines:

Three lines a_{1}x + b_{1}y + c_{1} = 0 a_{2}x + b_{2}y + c_{2} = 0 and a_{3}x + b_{3}y + c_{3} = 0 are concurrent if \(\left|\begin{array}{lll}

a_1 & b_1 & c_1 \\

a_2 & b_2 & c_2 \\

a_3 & b_3 & c_3

\end{array}\right|\) = 0

**Family Of Lines:**

(i) Equation of lines parallel to the line ax + by + c = 0 is given by: ax + by + λ = 0

(ii) Equation of lines perpendicular to the line ax + by + c = 0 is given by bx – ay + λ = 0

(iii) Equation of lines passing through the point of intersection of two lines.

a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 is given by: (a_{1}x + b_{1}y + c_{1}) + λ(a_{2}x + b_{2}y + c_{2})

Distance of a point from a line:

The perpendicular distance of A(x_{1}, y_{1}) from the line ax + by + c = 0 is: d = \(\left|\frac{a x_1+b y_1+c}{\sqrt{a^2+b^2}}\right|\)

Distance between two parallel lines:

ax + by + c_{1} = 0 and ax + by + c_{2} = 0 is d = \(\left|\frac{c_1-c_2}{\sqrt{a^2+b^2}}\right|\)

Position of a point with respect to a line:

A point A(x_{1}, y_{1}) lies

(i) above the line ax + by + c = 0 if \(\frac{a x_1+b y_1+c}{b}\) > 0

(ii) below the line ax + by + c = 0 if \(\frac{a x_1+b y_1+c}{b}\) < 0

Equation of bisectors of angle between two intersecting lines:

(i) Equation of angle bisector of two lines. a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 is given by \(\frac{a_1 x+b_1 y+c_1}{\sqrt{a_1^2+b_1^2}}=\pm \frac{a_2 x+b_2 y+c_2}{\sqrt{a_2^2+b_2^2}}\)

Note:

Out of two bisector take one and find the angle between that bisector and one line. If the angle is less than 45° then that bisector is the bisector of acute angle, otherwise, the other bisector is the bisector of acute angle.

(ii) Bisector of angle containing a given point (h, k):

Step – 1: Check the sign of a_{1}h + b_{1}k + c_{1} and a_{2}h + b_{2}k + c_{2}

- If they have same sign then the bisector of angle containing (h, k) is: \(\frac{a_1 x+b_1 y+c_1}{\sqrt{a_1^2+b_1^2}}=\frac{a_2 x+b_2 y+c_2}{\sqrt{a_2^2+b_2^2}}\)
- If they have opposite sign then the bisector of angle containing (h, k) is: \(\frac{a_1 x+b_1 y+c_1}{\sqrt{a_1^2+b_1^2}}=-\frac{a_2 x+b_2 y+c_2}{\sqrt{a_2^2+b_2^2}}\)

**Change Of Axes (Shifting Of Origin):**

(i) Translation of coordinate axes.

Let O'(h, k) is the origin of system S’ with respect to origin O(0, 0) of the system S. S’ is the translation of S. If (x, y) and (x’, y’) are the coordinate of a point P in the system S and S’ respectively then

x’ = x – h and y’ = y – k Or, x = x’ + h, y = y’ + k

(ii) Rotation of axes:

Let S’ is a rotation of S, α is the measure of rotation

If (x, y) and (x’, y’) are the coordinate of a point P with respect to S and S’ then x = x’ cos α – y’ sin α and y = x’ sin α + y’ cos α

(iii) Translation as well as a rotation:

If S’ is a combination of translation followed by a rotation then x = h + x’ cos α – y’ sin α, y = k + x’ sin α + y’ cos α