Question

12.
A straight line passes through a point A(1, 2) and makes an angle 60º with the x axis. this
line intersects the line x + y = 6 at P. Then AP will be
(A) 3(root3+1) (B) 3(root3 - 1)
(C) (root3+1)
(D) (root3,root3)​

Answer 1

First we find the equation of the line passing through A(1, 2) and making an angle 60° with the x axis.

Here the slope of the line is tan 60° = √3.

So we have the equation with the point A(1, 2),

y - 2 = (x - 1)√3

x√3 - y = √3 - 2

Since this line intersects the line x + y = 6 at P, the coordinates of the point P satisfy both the equations,

x√3 - y = √3 - 2 → (1)

and

x + y = 6 → (2)

Adding (1) and (2), we get,

x = (√3 + 4) / (√3 + 1)

x = (3√3 - 1) / 2

And from (2),

y = (13 - 3√3) / 2

Hence the point P is P((3√3 - 1) / 2, (13 - 3√3) / 2).

Now let's find the length of AP.

AP = √{[((3√3 - 1) / 2) - 1]² + [((13 - 3√3) / 2) - 2]²}

AP = √{[(3/2)(√3 - 1)]² + [(3√3/2)(√3 - 1)]²}

AP = (√3 - 1)√{(9/4) + (27/4)}

AP = 3(√3 - 1)

Hence (B) is the answer.