find the POINT on the line 6x+7y-5=0 which is closest to the point (2,2)
Well, finding the point on a line closest to another point can be a real "point" of concern. Let's put on our mathematical noses and solve it!
The line is given by the equation 6x + 7y - 5 = 0, and we're looking for a point on this line that is closest to the point (2,2).
The distance between two points can be calculated using the distance formula: √((x₂ - x₁)² + (y₂ - y₁)²).
So, using this formula, we can calculate the distance between the given point (2,2) and any point on the line. However, to find the minimum distance, we need to find where this distance function is at its lowest.
Phew, it sounds like we're juggling a lot of calculations here. Let's simplify things by using a shortcut. The shortest distance between a point and a line occurs when the line is perpendicular to the line connecting the two points.
The line given by 6x + 7y - 5 = 0 can be rewritten as 7y = -6x + 5, and by rearranging, we get y = (-6/7)x + 5/7. The line perpendicular to this line will have the slope that is the negative reciprocal, which is 7/6 in this case.
Now, we solve these two equations to find the point of intersection, which will be the closest point to (2,2).
So, setting (-6/7)x + 5/7 = (7/6)x + b, we can solve for x, and then substitute the value of x back into either equation to find y.
Now, our calculations are complete, and we have found the lucky point of intersection. I hope my mathematically clownish explanation helped you!
To find the point on the line \(6x + 7y - 5 = 0\) that is closest to the point (2, 2), we can use the concept of perpendicular distance.
Step 1: Rewrite the equation of the line in slope-intercept form \(y = mx + c\).
We'll rearrange the given equation to solve for y:
\(6x + 7y - 5 = 0\)
\(7y = -6x + 5\)
\(y = \frac{-6}{7}x + \frac{5}{7}\)
Step 2: Find the slope of the given line.
Comparing the equation to \(y = mx + c\), we can see that the slope (m) is \(\frac{-6}{7}\).
Step 3: Find the negative reciprocal of the slope.
The negative reciprocal of \(\frac{-6}{7}\) is \(\frac{7}{6}\).
Step 4: Find the equation of the line perpendicular to the given line that passes through the point (2, 2).
Using the point-slope form, the equation of the line is:
\(y - y_1 = m_\perp(x - x_1)\), where \((x_1, y_1)\) is the given point and \(m_\perp\) is the negative reciprocal of the slope.
Substituting the values, we have:
\(y - 2 = \frac{7}{6}(x - 2)\)
Simplifying:
\(6(y - 2) = 7(x - 2)\)
\(6y - 12 = 7x - 14\)
\(7x - 6y = 2\)
Step 5: Solve the system of equations formed by the given line and the perpendicular line.
We need to solve the system of equations:
\(6x + 7y - 5 = 0\) and \(7x - 6y = 2\)
Solving this system of equations will give us the coordinates of the point that lies on both lines.
Step 6: Substitute the value of one variable obtained from the second equation into the first equation.
Let's solve the second equation for \(x\):
\(7x - 6y = 2\)
\(7x = 6y + 2\)
\(x = \frac{6}{7}y + \frac{2}{7}\)
Now, substitute this value of \(x\) into the first equation:
\(6\left(\frac{6}{7}y + \frac{2}{7}\right) + 7y - 5 = 0\)
Simplifying:
\(\frac{36}{7}y + \frac{12}{7} + 7y - 5 = 0\)
\(\frac{85}{7}y = \frac{23}{7}\)
\(y = \frac{23}{85}\)
Step 7: Substitute the value of \(y\) back into the equation obtained when solving for \(x\) to find the value of \(x\).
\(x = \frac{6}{7}\left(\frac{23}{85}\right) + \frac{2}{7}\)
Simplifying:
\(x = \frac{6}{7} \cdot \frac{23}{85} + \frac{2}{7}\)
\(x = \frac{6}{7} \cdot \frac{23}{85} + \frac{2 \cdot 85}{7 \cdot 85}\)
\(x = \frac{138}{595} + \frac{170}{595}\)
\(x = \frac{308}{595}\)
Step 8: The coordinates of the point on the line closest to the point (2, 2) are \(\left(\frac{308}{595}, \frac{23}{85}\right)\).
To find the point on the line 6x + 7y - 5 = 0 that is closest to the point (2, 2), we need to follow these steps:
Step 1: Understand the problem
The equation 6x + 7y - 5 = 0 represents a line in the coordinate plane. We want to find a point on this line that is closest to the point (2, 2).
Step 2: Determine the distance formula
To find the distance between two points in the coordinate plane, we use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the point (x1, y1) is given as (2, 2), and we will be comparing it to various points on the given line.
Step 3: Convert the given equation to slope-intercept form
First, we rearrange the equation 6x + 7y - 5 = 0 to the slope-intercept form (y = mx + b) by isolating y:
7y = -6x + 5
y = (-6/7)x + 5/7
Now we have the equation of the line in slope-intercept form: y = (-6/7)x + 5/7
Step 4: Calculate the distance between (2, 2) and any point on the line
Since we want to find the point on the line that is closest to (2, 2), we substitute (-6/7)x + 5/7 for y in the distance formula and solve for x.
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((x - 2)^2 + ((-6/7)x + 5/7 - 2)^2)
Step 5: Minimize the distance
To minimize the distance d, we differentiate d with respect to x, set it equal to zero, and solve for x:
d' = 0
By solving this equation, we will find the x-coordinate of the point on the line closest to (2, 2).
Step 6: Find the y-coordinate
Once we have the x-coordinate, substitute it back into the equation y = (-6/7)x + 5/7 to calculate the corresponding y-coordinate.
Step 7: Calculate the final point
Using the x-coordinate and y-coordinate obtained in the previous steps, we can determine the coordinates of the point on the line that is closest to (2, 2).