Question

Find the equation of a line which divides the line segment joining the points \((1,1)\) and \((2,3)\) in the ratio \(2: 3\) perpendicularly. (1) \(5 x-5 y+2=0\) (2) \(5 x+5 y+2=0\) (3) \(x+2 y-5=0\) (4) \(x+2 y+7=0\)

Short answer

Provide a short answer based on the given solution: The equation of the line dividing the segment joining (1, 1) and (2, 3) in the ratio 2:3 perpendicularly is 5x + 10y - 25 = 0.

Step-by-step solution

Find the point where line divides the segment in the given ratio

To find the point dividing the line segment in ratio 2:3, we use the formula: \(P(x, y) = \left (\frac{m_1x_2 + m_2x_1}{m_1+m_2}, \frac{m_1y_2+m_2y_1}{m_1+m_2} \right ) \) where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the given points and \(m_1\) and \(m_2\) are the values, forming the ratio. Our points are \((1, 1)\) and \((2, 3)\), and the ratio is 2:3, meaning \(m_1=2\) and \(m_2=3\). Thus, \(P(x, y) = \left(\frac{2 \cdot 2 + 3\cdot 1}{2+3}, \frac{2\cdot 3 + 3\cdot 1}{2+3} \right )\) \( = \left( \frac{7}{5}, \frac{9}{5} \right ) \)

Calculate the slope of the given line segment

The slope of the line segment connecting \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) So in this case, the slope of the line segment is: \(m = \frac{3-1}{2-1} = \frac{2}{1} = 2\)

Find the slope of the perpendicular line

To find the slope of the perpendicular line, we take the negative reciprocal of the given line segment's slope. So, \(m_{perpendicular} = -\frac{1}{2}\)

Write the equation of the line

Now, we have the slope of the perpendicular line as \(-\frac{1}{2}\) and also the point, \(\left(\frac{7}{5}, \frac{9}{5}\right)\), through which it passes. We use the point-slope form to find the equation of the line: \(y - y_1 = m_{perpendicular}(x - x_1)\) Substituting the values, we get: \(y - \frac{9}{5} = -\frac{1}{2}(x - \frac{7}{5})\) To get rid of the fractions, we multiply both sides by \(10\), obtaining: \(10y - 18 = -5(x - \frac{7}{5})\) Then, we distribute the \(-5\) on the right side: \(10y - 18 = -5x + 7\) Rearrange the equation to bring it to the correct format, we have: \(5x + 10y - 25 = 0\) So, the equation of the line dividing the segment joining \((1, 1)\) and \((2, 3)\) in the ratio \(2:3\) perpendicularly is as follows: \(5x + 10y - 25 = 0\)

Key concepts

Equation of a Line

The equation of a line is fundamental in coordinate geometry. It represents a straight path extending in both opposing directions without end. Generally, the equation of a line can be expressed in various forms, with the most common being the slope-intercept form, point-slope form, and standard form. Each form provides unique insights into the properties of the line, making it easier to understand and solve geometry problems. In this exercise, we sought a perpendicular line dividing a segment, leading us to use the point-slope form to express it based on known slopes and coordinates.

Point-Slope Form

The point-slope form is particularly useful when you have a point on the line and the slope. The equation is represented as \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. This form quickens the process of finding the linear equation because it directly incorporates the slope and a specific point. In our particular exercise, we tapped into the point-slope form to derive the equation of a line passing through the point \( \left(\frac{7}{5}, \frac{9}{5}\right)\) with a slope of \(-\frac{1}{2}\), providing a straightforward approach to writing out the line's equation.

Slope Calculation

Calculating the slope is crucial in understanding the direction and steepness of a line. It is calculated using the formula \( m = \frac{y_2-y_1}{x_2-x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on a line. The slope indicates how much the line rises or falls as it moves from left to right. In our scenario, we calculated the slope between points \((1, 1)\) and \((2, 3)\), finding it to be 2. Subsequently, we determined the slope of the perpendicular line as \(-\frac{1}{2}\), which provides the necessary conditions for perpendicularity in lines, being the negative reciprocal of the original slope.

Line Segment Division

Line segment division involves dividing a line segment into two parts at a specific ratio. This concept is essential when determining points that divide segments in various applications across mathematics. The formula \( P(x, y) = \left( \frac{m_1x_2 + m_2x_1}{m_1+m_2}, \frac{m_1y_2 + m_2y_1}{m_1+m_2} \right) \) helps find such a point, ensuring precision and accuracy. In the given task, we applied this formula for the line segment between points \((1, 1)\) and \((2, 3)\) using a ratio of \(2:3\), finding the point \(\left(\frac{7}{5}, \frac{9}{5}\right)\). Understanding how to effectively apply this formula ensures accurate solutions in practical and theoretical geometry problems.