Question: A and B are the points (2,0) and (0,2) respectively. The coordinates of the point P on the line 2x+3...

Question

Answer 1

Solution

Hint: To solve this problem, we can use triangle inequality which states that the sum of two sides of a triangle is always greater than the third side. Also, the modulus of the difference between two sides is always less than the third side.

Complete step by step answer:
Let us assume that the point P(x1, y1) lies on 2x+3y+1=0. We have been given that
A(2, 0) and B(0, 2)
By the triangle inequality, the maximum value of ∣PA−PB∣ is equal to the AB
Consider PAB is triangle then from triangular inequalities-
PB + AB > PA
∣PA−PB∣ < AB
Now we can write that-
$\mathrm{AB}=\sqrt{\left(0-2\right)^2+\left(2-0\right)^2}\\\\\mathrm{AB}=\sqrt8\\\\\mathrm{AB}=2\sqrt2$
$2\sqrt2>\left|\mathrm{PA}-\mathrm{PB}\right|\;\mathrm{is}\;\mathrm{the}\;\mathrm{same}\;\mathrm{as}-\\\2\sqrt2>\mathrm{equation}\;\mathrm{of}\;\mathrm{line}\;\mathrm{AB}\\\$
on solving above equation we get
y1 = −1(x1 − 2)
y1 = −x1 + 2
x1 + y1 = 2
Point P lies in line 2x+3y=1
2x1 + 3y1 = −1
Solving x1 and y1 using the two equations-
x1 = 7, y1 = −5
Point P(7, −5)
Hence, when |PA-PB| is maximum, the point P is(7, -5).

The correct option is A. (7,-5) if |PA-PB| is maximum

Note: We can also do this question by a very long method. First, we will assume the value of point P in parametric form, then find PA and PB. Then we will differentiate the function |PA-PB| to find maxima and minima and hence find the value of point P.