If a straight line through the point P(1,2) , which makes an angle 45^ with the X -axis, meets the line 3 x+4 y+5=0 at Q , then length of P Q equals ......... units

Step-by-step solution

Equation of line passing through the point P(1,2) which makes an angle 45^ with the X -axis is aligned (y-2)=m(x-1) and m & = & = 45^ =1 aligned Equation is y-2=1(x-1) y-2=x-1 x-y+1=0 Point of intersection of lines x-y+1=0 and 3 x+4 y+5=0 is Q . Then, for Q x-y+1=0, y=x+1 Put it in 3 x+4 y+5=0 aligned & 3 x+4(x+1)+5=0 & 3 x+4 x+4+5=0 7 x+9=0 & x=- 9 7 , y=- 9 7 +1=- 2 7 & Q is (- 9 7 ,- 2 7 ) & P Q= (x₂-x₁ )^2+ (y₂-y₁ )^2 [distance formula] & = (1+ 9 7 )^2+ (2+ 2 7 )^2 = ( 16 7 )^2+ ( 16 7 )^2 & = 2 ( 16 7 )^2 = 16