Class 11 Maths Chapter 10

Perpendicular Line Equation

• A line is perpendicular to another if the product of their slopes equals -1.
• The equation $3x - 4y + 7 = 0$ can be rearranged to slope-intercept form: $y = \frac{3}{4}x + \frac{7}{4}$, revealing a slope of $\frac{3}{4}$.
• The slope of the perpendicular line is $-\frac{4}{3}$.
• Using point-slope form $y - y_1 = m(x - x_1)$, where $(x_1, y_1) = (2, 5)$, the equation is: $y - 5 = -\frac{4}{3}(x - 2)$.

Distance Between Parallel Lines

• Parallel lines of the form $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ have a distance formula given by: $d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}$.
• For lines $2x - 3y + 5 = 0$ and $2x - 3y + 7 = 0$, constants are $C_1 = 5$ and $C_2 = 7$.
• The distance is calculated as $d = \frac{|7 - 5|}{\sqrt{2^2 + (-3)^2}} = \frac{2}{\sqrt{13}}$.

Angle Between Lines

• The angle $\theta$ between two lines can be found using the formula: $\tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right|$.
• The slopes can be derived from the given line equations.
• For lines $2x - 3y + 5 = 0$ (slope $\frac{2}{3}$) and $4x - 6y + 7 = 0$ (slope $\frac{2}{3}$), since these lines are parallel, the angle is $0°$.

Angle Bisector Equation

• The angle bisector of lines $Ax + By + C_1 = 0$ and $Dx + Ey + C_2 = 0$ can be derived using the formula involving the coefficients of the equations.
• For lines $3x - 4y + 7 = 0$ and $4x - 3y + 5 = 0$, apply the bisector angle formula.
• The resulting equation is: $\frac{3x - 4y + 7}{\sqrt{3^2 + (-4)^2}} = \pm \frac{4x - 3y + 5}{\sqrt{4^2 + (-3)^2}}$.

Line at 45° Angle

• A line intersecting another at a $45°$ angle implies that the slope of the new line can be $\frac{3}{4} + 1$ or $\frac{3}{4} - 1$.
• With the line $3x - 4y + 7 = 0$ rearranged for its slope, use the point-slope form with point (1, 2).
• Solve for either desired slope case to find the required line equation.

Test your knowledge of straight lines with these multiple-choice questions for Class 11 Maths Chapter 10. Access the MCQs online to prepare for the CBSE examination and improve your understanding of the topic.