From any point P on the line x = 2y perpendicular is drawn on y = x. Let foot of perpendicular is Q. Find the locus of mid point of PQ.a)2x = 3y b)5x = 7yc)3x = 2y d)7x = 5yCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

slope of PQ =

= -1
⇒ k – α = - h + 2α
⇒ α =

......... (1)
Also 2h = 2α + β
2k = α + β
⇒ 2h = α + 2k
⇒ a = 2h - 2k .....(2)
from (1) & (2)

= 2(h - k)
so locus is 6x –6y = x + y
⇒ 5x = 7y

Answer

Can you explain this question in Hindi or Gujarati so I give you correct formula for this question.

Answer

Given:
- The line x = 2y
- A perpendicular is drawn from any point P on the line to the line y = x
- Let Q be the foot of the perpendicular
- We need to find the locus of the midpoint of PQ

Approach:
1. Find the equation of the line y = x in terms of y.
2. Find the coordinates of point Q using the given information.
3. Find the midpoint M of PQ.
4. Write the coordinates of M in terms of x and y.
5. Simplify the equation to obtain the locus of M.

Solution:

Step 1: Find the equation of the line y = x in terms of y.
- The equation of the line y = x can be rewritten as x - y = 0.

Step 2: Find the coordinates of point Q using the given information.
- Since the line x = 2y is perpendicular to the line y = x, the slope of x = 2y is the negative reciprocal of the slope of y = x.
- The slope of y = x is 1, so the slope of x = 2y is -1/1 = -1.
- The line x = 2y can be rewritten as y = (1/2)x.
- The slope-intercept form of the line y = (1/2)x is y = mx + b, where m is the slope and b is the y-intercept.
- Comparing the equations, we can see that m = 1/2 and b = 0.
- So, the equation of the line x = 2y in slope-intercept form is y = (1/2)x + 0.
- The slope-intercept form of the line gives us the coordinates of point Q as (2t, t), where t is a parameter.

Step 3: Find the midpoint M of PQ.
- Let the coordinates of point P be (x, y).
- Using the equation of the line x = 2y, we can substitute x = 2y into the equation x - y = 0 to find the coordinates of point Q.
- Substituting x = 2y into x - y = 0, we get 2y - y = 0.
- This simplifies to y = 0, which means point Q is (0, 0).
- The coordinates of point P are (x, y) and the coordinates of point Q are (0, 0).
- The midpoint M of PQ can be found using the formula (x1 + x2)/2, (y1 + y2)/2.
- Substituting the coordinates of P and Q, we get the midpoint M as (x/2, y/2).

Step 4: Write the coordinates of M in terms of x and y.
- The coordinates of M are (x/2, y/2).
- Rewriting this in terms of x and y, we get (1/2)x, (1/2)y.

Step 5: Simplify the equation to obtain the locus of M.
- The locus of M can be obtained by eliminating the parameter t from the coordinates (1/2)(2t), (

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