Question

The point P(a,b) lies on the line through A(-1,-2) and B(3,0) and PA=`125`{the square root of 125}. what are the values of a and b?

PA= `125`

Answer 1

`a=9, b=3\ \ or \ \ a=-11, b=-7`

Explanation:

The slope of line AB passing through the points `A(-1, -2)` & `B(3, 0)` will be equal to that of line AB passing through the points `P(a, b)` & `A(-1, -2)` because the line is the same

`\frac{b-(-2)}{a-(-1)}=\frac{-2-0}{-1-3}`

`a=2b+3\ ...........(1)`

Now, suing distance formula, the distance between the points `P(a, b)` & `A(-1, -2)` is given as

`PA=\sqrt{(a-(-1))^2+(b-(-2))^2}`

`\sqrt{(a+1)^2+(b+2)^2}=\sqrt125\quad (\because PA=\sqrt125)`

`(a+1)^2+(b+2)^2=125`

`(2b+3+1)^2+(b+2)^2=125`

`5(b+2)^2=125`

`b+2=\pm 5`

`b=3, -7`

setting the above values of `b` in (1), we get corresponding values of `a` as follows

`a=2(-2\pm5)+3`

`a=9, -11`

Hence, we get

`a=9, b=3\ \ or \ \ a=-11, b=-7`