two points A and B are given. find the set of feet of the perpendicular dropped from the point A onto all possible straight lines passing through the point B
To find the set of feet of the perpendicular dropped from point A onto all possible straight lines passing through point B, follow these steps:
1. Calculate the slope of the line passing through points A and B.
- The slope formula is given by: m = (yb - ya) / (xb - xa), where (xa, ya) and (xb, yb) are the coordinates of points A and B, respectively.
2. Find the negative reciprocal of the slope obtained in step 1. Let's call it m_perp.
- The negative reciprocal of a slope is obtained by flipping the fraction and negating it. For example, if the original slope is 2/3, the negative reciprocal would be -3/2.
3. Calculate the equation of the line passing through point A and having slope m_perp. Let's call this line L1.
- The equation of a line in point-slope form is given by: y - ya = m_perp * (x - xa), where (xa, ya) are the coordinates of point A.
- Rearrange the equation to slope-intercept form (y = mx + b) to get the equation of L1.
4. Find the equation of the line passing through point B, perpendicular to line L1. Let's call this line L2.
- The slope of L2 will be the negative reciprocal of m_perp since the two lines are perpendicular.
- Calculate the equation of L2 using the same process as in step 3.
5. The set of feet of the perpendiculars dropped from point A onto all possible lines passing through point B is the intersection of line L1 and line L2.
- Find the coordinates of the intersection point using simultaneous equations of L1 and L2.
- These coordinates represent the feet of the perpendiculars onto all possible lines passing through point B.
To find the set of feet of the perpendicular dropped from point A onto all possible straight lines passing through point B, we can follow these steps:
1. Determine the equation of the line passing through points A and B.
- Given two points A(x1, y1) and B(x2, y2), the equation of the line passing through them can be found using the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
- Calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1).
- Substitute the slope and one of the points (x1, y1) in the equation to find the y-intercept (b).
- The equation of the line passing through points A and B is now determined.
2. Find the equation of the perpendicular line to the line obtained in step 1.
- Perpendicular lines have negative reciprocal slopes. So, if the slope of the line obtained in step 1 is m, the slope of the perpendicular line would be -1/m.
- Using the point-slope form, substitute the slope (-1/m) and one of the points (x1, y1) to obtain the equation of the perpendicular line.
3. Find the intersection point between the line from step 1 and the perpendicular line from step 2.
- To find the intersection point, we equate the equations of the two lines and solve for the coordinates (x, y).
4. Repeat steps 1-3 for all possible straight lines passing through point B.
- Consider different slopes for the lines passing through point B and calculate the equations and intersection points with the perpendicular line in each case.
5. The set of feet of the perpendicular dropped from point A onto all possible straight lines passing through point B is the collection of intersection points obtained in step 4.