Arun
Last Activity: 6 Years ago
Solving the equation given for y:
4x + 3y + 9 = 0
==> 3y = -4x - 9
==> y = (-4/3)x - 3.
So any point on this line takes the form of [x, (-4/3)x - 3].
The distance between A(0, 1) and P[x, (-4/3)x - 3] is:
d(P, A) = √{[(-4/3)x - 3 - 1]^2 + (x - 0)^2]} = √[(25/9)x^2 + (32/3)x + 16].
The distance between B(2, 0) and P[x, (-4/3)x - 3] is:
d(P, B) = √{[(-4/3)x - 3]^2 + (x - 2)^2} = √[(25/9)x^2 + 4x + 13].
Then:
d(P, A) - d(P, B) = √[(25/9)x^2 + (32/3)x + 16] - √[(25/9)x^2 + 4x + 13].
Taking derivatives of √[(25/9)x^2 + (32/3)x + 16] - √[(25/9)x^2 + 4x + 13], we see that:
d/dx √[(25/9)x^2 + (32/3)x + 16] - √[(25/9)x^2 + 4x + 13]
= [(50/9)x + 32/3]/{2√[(25/9)x^2 + (32/3)x + 16]} - [(50/9)x^2 + 4]/{2√[(25/9)x^2 + 4x + 13]}.
Setting this expression equal to zero yields x = -24/5. This is the x-coordinate of P. Then, the corresponding y-coordinate is (-4/3)(-24/5) - 3 = 17/5. Therefore, the required point is (-24/5, 17/5) as required.
