Question

(Problem in two-dimensions.) Given the line: L1 : y = 2x , (2) find the equation of the line L2 perpendicular to L1 passing through the point P = (1, 2).

Answer 1

Given :

Equation of line 1 , y = 2x .

A point (1 , 2) .

To Find :

The equation of the line L2 perpendicular to L1 passing through the point P = (1, 2) .

Solution :

Let , equation of line 2 is :

y = mx + c     .....eq 1 ( here , m is slope and c is a constant )

Now , we know when two lines are perpendicular product of their slope is -1 .

Slope of given line is 2 .

Therefore ,

$2m=-1\\\\m=\dfrac{-1}{2}$

Now putting value of m in equation 1 , we get :

$y=\dfrac{-1}{2}x+c$

Now , it is given that this point (1,2) satisfy the above equation .

So ,

$2=\dfrac{-1}{2}(1)+c\\\\c=\dfrac{5}{2}$

Putting value of c in above equation , we get :

$y=-\dfrac{1}{2}x+\dfrac{5}{2}\\\\2y=-x+5$

Therefore , the equation of the line L2 perpendicular to L1 passing through the point P = (1, 2) is 2y = -x +5 .

Hence , this is the required solution.