
Find the foot of perpendicular and image of point along the line .
Answer
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Hint: Here,we will draw a figure representing this situation. Then, we will find the direction ratios of the line. As both the lines are perpendicular to each other, we will be able to find the value of . Using this we will find the required coordinates of the foot of the perpendicular which could be used as a midpoint to find the coordinates of the image of the given point.
Formula Used:
We will use the following formulas:
1. If an equation of a line is given by .
Then, represents the direction ratios of the given line
2. The direction ratios of a line passing through two points and is given by .
3. If two lines are perpendicular to each other, then, the sum of product of their direction ratios is 0.
4. Coordinates of the mid-point is given as .
Complete step-by-step answer:
Given equation of line is
Now, we will cross multiply to find the respective values of
Hence, we get,
Now, the direction ratios of the given line are: because if an equation of a line is given by
Then, represents the direction ratios of the given line
Now, we will draw a figure representing the given situation.
Now, let the coordinates of the foot of perpendicular line of the given line be
Since this foot of perpendicular lies on both the lines as it is their point of intersection.
Thus, it will satisfy the given equation of the line.
Hence, substituting in the equation of the line and after cross multiplying, we get,
………………………………….
Now, the direction ratios of a line passing through two points and is given by
Since, the perpendicular line consists of the given point and it also passes through the foot of perpendicular
Hence, the direction ratios of the perpendicular line are:
Now, using the fact that if two lines are perpendicular to each other, then, the sum of product of their direction ratios is 0, we get,
Here, substituting the values of from , we get,
Solving further, we get,
Dividing both sides by 2
Adding 2 on both sides
And, dividing both sides by 7, we get,
Substituting this value in , we get,
Therefore, the coordinates of foot of perpendicular are
Now, if the coordinates of the image of the given point is
Then, we know that the foot of perpendicular should be the midpoint of the perpendicular line .
Hence, coordinates of the mid-point will be:
Here, substituting and
Coordinates of the mid-point will be
But, the coordinates of foot of perpendicular
Hence, we get,
After comparing, we get,
Subtracting 7 from both sides, we get
Dividing both sides by 7, we get
Similarly,
Subtracting 14 from both sides, we get
Dividing both sides by 7, we get
Also,
Subtracting 7 from both sides, we get
Dividing both sides by 7, we get
Therefore, the coordinates of the image of the given point
Hence, the foot of perpendicular and image of point along the line are and respectively.
Thus, this is the required answer.
Note:
When two lines meet or intersect each other at right angles or 90 degrees, they are said to be perpendicular to each other. Now, when a point is being reflected on a mirror or a line then, the distance of the point from the line is equal to the distance of its image behind the line. Thus, the point of intersection or the foot of perpendicular turns out to be the midpoint of the line. Hence, these facts about reflection should be kept in mind while solving these types of questions.
Formula Used:
We will use the following formulas:
1. If an equation of a line is given by
Then,
2. The direction ratios of a line passing through two points
3. If two lines are perpendicular to each other, then, the sum of product of their direction ratios is 0.
4. Coordinates of the mid-point
Complete step-by-step answer:
Given equation of line is
Now, we will cross multiply to find the respective values of
Hence, we get,
Now, the direction ratios of the given line are:
Then,
Now, we will draw a figure representing the given situation.

Now, let the coordinates of the foot of perpendicular line of the given line be
Since this foot of perpendicular lies on both the lines as it is their point of intersection.
Thus, it will satisfy the given equation of the line.
Hence, substituting
Now, the direction ratios of a line passing through two points
Since, the perpendicular line consists of the given point
Hence, the direction ratios of the perpendicular line are:
Now, using the fact that if two lines are perpendicular to each other, then, the sum of product of their direction ratios is 0, we get,
Here, substituting the values of
Solving further, we get,
Dividing both sides by 2
Adding 2 on both sides
And, dividing both sides by 7, we get,
Substituting this value in
Therefore, the coordinates of foot of perpendicular are
Now, if the coordinates of the image of the given point is
Then, we know that the foot of perpendicular should be the midpoint of the perpendicular line
Hence, coordinates of the mid-point
Here, substituting
But, the coordinates of foot of perpendicular
Hence, we get,
After comparing, we get,
Subtracting 7 from both sides, we get
Dividing both sides by 7, we get
Similarly,
Subtracting 14 from both sides, we get
Dividing both sides by 7, we get
Also,
Subtracting 7 from both sides, we get
Dividing both sides by 7, we get
Therefore, the coordinates of the image of the given point
Hence, the foot of perpendicular and image of point
Thus, this is the required answer.
Note:
When two lines meet or intersect each other at right angles or 90 degrees, they are said to be perpendicular to each other. Now, when a point is being reflected on a mirror or a line then, the distance of the point from the line is equal to the distance of its image behind the line. Thus, the point of intersection or the foot of perpendicular turns out to be the midpoint of the line. Hence, these facts about reflection should be kept in mind while solving these types of questions.
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