04_Ex 4.4

Exercise Questions

Questions	Links
Q1. Find the point of intersection…	4.4 Q-1
Q2. Find an equation of the line through…	4.4 Q-2
Q3. Find an equation of the line through…	4.4 Q-3
Q4. Find the condition that the lines…	4.4 Q-4
Q5. Determine the value of $p$ such that…	4.4 Q-5
Q6. Show that the lines…	4.4 Q-6
Q7. The vertices of a triangle are…	4.4 Q-7
Q8. Check whether the lines…	4.4 Q-8
Q9. Find the coordinates of the triangle formed…	4.4 Q-9
Q10. Find the angle measured from $\ell_{1}$ to…	4.4 Q-10
Q11. Find the interior angles of the triangle…	4.4 Q-11
Q12. Find the interior angles of the quadrilateral…	4.4 Q-12
Q13. Show that the points $A(-1,-1), B(-3,0)$…	4.4 Q-13
Q14. Find the area of region bounded by…	4.4 Q-14
Q15. The vertices of a triangle are…	4.4 Q-15
Q16. Expression the given system of equation in…	4.4 Q-16
Q17. Find a system of linear equations…	4.4 Q-17

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Overview

This exercise focuses on fundamental concepts of line equations, slopes, concurrency, and their applications in coordinate geometry. Understanding these concepts is crucial for solving problems related to line intersections, equations through specific points, and geometric properties.

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Key Concepts

1. Line Equation: The general form of a line in 2D space is:

$$Ax + By + C = 0$$

Here, $A$, $B$, and $C$ are constants.

2. Slope (m): Defined as the ratio of the change in $y$ to the change in $x$, calculated as:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

3. Concurrent Lines: Lines are concurrent if they intersect at a single point. For lines $l_1$, $l_2$, and $l_3$, concurrency condition is:

$$\left|\begin{array}{ccc} a_1 & b_1 & c_1 a_2 & b_2 & c_2 a_3 & b_3 & c_3 \end{array}\right| = 0$$

4. Point of Intersection: The coordinates of the intersection of two lines $l_1$ and $l_2$ can be determined using determinants:

$$x = \frac{\left|\begin{array}{cc} B_1 & C_1 B_2 & C_2 \end{array}\right|}{\left|\begin{array}{cc} A_1 & B_1 A_2 & B_2 \end{array}\right|}, \quad y = \frac{\left|\begin{array}{cc} A_1 & C_1 A_2 & C_2 \end{array}\right|}{\left|\begin{array}{cc} A_1 & B_1 A_2 & B_2 \end{array}\right|}$$

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Important Formulas

1. Line through Two Points:

$$y - y_1 = m(x - x_1)$$

where $m = \frac{y_2 - y_1}{x_2 - x_1}$.

2. Point of Intersection: See the determinant formulas under “Key Concepts.”

3. Equal Intercepts:

$$x + y = c$$

where $c$ is the intercept value.

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Tips and Tricks

1. Slope Calculation: Always ensure proper subtraction order to avoid sign errors.
2. Concurrency Test: Apply the determinant condition only if all lines are in standard form.
3. Visual Confirmation: Use graphing for quick verification of intersections.
4. Parallelism and Perpendicularity:
   • Lines are parallel if slopes are equal.
   • Lines are perpendicular if the product of their slopes equals $-1$.

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Summary

This exercise solidifies understanding of line equations, slopes, and concurrency. It focuses on analytical techniques for finding intersections and line equations, essential for advanced geometry and problem-solving.

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Reference

By Sir Shahzad Sair:

By Great Science Academy

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