Question

The foot of perpendicular drawn from \((2,3)\) to the line \(4 x-5 y-34=0\) is ..... (a) \((6,-2)\) (b) \([(246 / 41),(82 / 41)]\) (c) \((-6,2)\) (d) None of these

Short answer

The foot of the perpendicular drawn from \((2,3)\) to the line \(4x-5y-34=0\) is (d) None of these.

Step-by-step solution

Identify the given point and the equation of the line

The given point is \((2, 3)\), and the equation of the line is \(4x - 5y - 34 = 0\).

Write down the formula for the foot of the perpendicular

We will use the following formula to find the foot of the perpendicular from the point \((x_{1}, y_{1})\) to the line \(Ax + By + C = 0\): Foot of the perpendicular => \( \left( x_{2}, y_{2} \right) = \left( \frac{ x_{1} - A \left( Ax_{1} + By_{1} + C \right) }{ A^2 + B^2 } , \frac{ y_{1} - B \left( Ax_{1} + By_{1} + C \right) }{ A^2 + B^2 } \right) \)

Substitute the given values into the formula

Let's substitute the given point and the coefficients of the line into the formula: \( \left( x_{2}, y_{2} \right) = \left( \frac{ 2 - 4 \left( 4(2) - 5(3) - 34 \right) }{ 4^2 + (-5)^2 } , \frac{ 3 - (-5) \left( 4(2) - 5(3) - 34 \right) }{ 4^2 + (-5)^2 } \right) \)

Simplify the expression

Now, let's simplify the expression step-by-step: \( \left( x_{2}, y_{2} \right) = \left( \frac{ 2 - 4 \left( 8 - 15 - 34 \right) }{ 16 + 25 } , \frac{ 3 - (-5) \left( 8 - 15 - 34 \right) }{ 16 + 25 } \right) \) \( \left( x_{2}, y_{2} \right) = \left( \frac{ 2 + 164 }{ 41 } , \frac{ 3 - 82 }{ 41 } \right) \) \( \left( x_{2}, y_{2} \right) = \left( \frac{ 166 }{ 41 } , \frac{ -79 }{ 41 } \right) \)

Compare the result with the given options

Now, let's compare the calculated foot of the perpendicular with the given options: (a) \((6,-2)\) (b) \([(246 / 41),(82 / 41)]\) (c) \((-6,2)\) (d) None of these None of the given options match the calculated result. Therefore, the correct answer is (d) None of these.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves the study of geometric figures through the use of coordinates and the principles of algebra. The basic elements include points, which are represented by pairs of numbers \( (x, y) \) corresponding to their position on the Cartesian plane. These coordinates allow us to represent lines, shapes, and curves, and to solve geometric problems algebraically.

In the context of our problem, the given point \( (2, 3) \) and the line \( 4x - 5y - 34 = 0 \) can be visualized on the Cartesian plane. By using coordinate geometry, we can work out the exact location where a perpendicular from the point meets the line. This application of coordinate geometry isn't just about plotting points but also involves using algebraic formulas to find solutions to geometric problems.

Perpendicular Distance

In geometry, the perpendicular distance from a point to a line is the shortest distance between them. This distance is represented by a line segment that is perpendicular to the line in question and passes through the given point. Finding the perpendicular distance has practical applications in various fields such as engineering and computer graphics.

To solve for this in coordinate geometry, a formula based on the coefficients of the line's equation and the coordinates of the point is used. This process was demonstrated in our exercise, where we calculated the foot of the perpendicular - the point at which the perpendicular distance touches the line - using the given equation and point. It's worth noting that getting this foot of the perpendicular right is critical, as small miscalculations can drastically alter the outcome, which is evident in the provided options versus the calculated result.

Equation of a Line

The equation of a line is a fundamental concept in coordinate geometry, as it provides a clear algebraic expression that describes all the points which constitute the line. The general form can be expressed as \( Ax + By + C = 0 \) where \( A \) and \( B \) are coefficients that determine the slope of the line, and \( C \) determines its position relative to the origin.

In the exercise, we focused on the perpendicular relationship between a point and a line, and the equation of the line played a crucial role in determining the foot of the perpendicular. Correctly identifying and manipulating these coefficients allowed us to use a specific formula to solve for the foot of the perpendicular, illustrating how the equation of a line is not just an abstract representation but a practical tool for problem-solving in geometry.