Question

The distance of the point p(4,1) from the line 4x-y=0 measured along the line making an angle of 135degree with the positive direction of x-axis is

Answer 1

Solution :-

Here we're given with coordinates of a point P (4 , 1) from which an angle is formed of 135° in the x-axis , and a line whose equation is given to us is intersecting that line from point P.

So we would be using the distance form of a line theorum :

• (x - x1 / cos θ) = (y- y1 / sin θ) = r

We have :

• x1 = 4
• y1 = 1
• θ = 135°

Substituting the values :

>> (x - 4 / cos 135°) = (y - 1 / sin 135°) = r

(Value of cos 135° is -1/√2 and sin 135° is 1/√2)

>> [x - 4 / (-1/√2) ] = [y - 1 / (1/√2) ] = r

Now,

>> -√2 (x - 4) = r

>> (x - 4) = - r / √2

>> x = 4 - (r/√2)

And,

>> √2 (y - 1) = r

>> (y - 1) = r / √2

>> y = 1 + r / √2

Here we got our x and y coordinates of that point where the line having point P was intersecting.

Assuming that point as A.

As we know that point A lies on line "4x - y = 0". So let's substitute the coordinates into it nd find out the value of r that is distance between A and P.

>> 4 (4 - r / √2) - (1 + r / √2) = 0

>> 16 - (4r / √2) - 1 - (r / √2) = 0

>> 15 - (4r / √2) - (r / √2) = 0

>> r = 3√2