Question

D. Read and analyze the problem below. Solve and show complete solution.

Two stations, located at M(1.5, 0) and N(1.5, 0) (units are in km), simultaneously send sound

signals to a ship, with the signal traveling at the speed of 0.33 km/s. If the signal from N was

received by the ship four seconds before the signal it received from M, find the equation of the

curve containing the possible location of the ship.

Answer 1

$\text {let us choose a coordinate system so that the X axis}$

$\text{ passes through the stations }M,N$

$\text{and the Y axis through the middle of the segment} MN$

$\text {let } MN= 2c\text{ then}$

$M(-c,0);N(-c;0)$

$S(x,y) - \text{coordinates of points satisfying the condition:}$

$SM - SN = 2a$

$2a -\text {this is the distance traveled by the sound for the difference in time}$

$\text{of signal fixation by the receiving stations}M,N$

$SM =\sqrt{(x+c)^2+y^2}$

$SN =\sqrt{(x-c)^2+y^2}$

$SM>SN$

$\sqrt{(x+c)^2+y^2}>\sqrt{(x-c)^2+y^2}$

$(x+c)^2+y^2>(x-c)^2+y^2$

$(x+c)^2>(x-c)^2$

$x^2+2xc+c^2>x^2-2xc+c^2$

$4xc>0$

$x>0(1)$

$SM - SN = 2a$

$\sqrt{(x+c)^2+y^2} - \sqrt{(x-c)^2+y^2} =2a(2)$

$\sqrt{(x+c)^2+y^2} =2a+\sqrt{(x-c)^2+y^2}$

$(x+c)^2+y^2 =4a^2+4a\sqrt{(x-c)^2+y^2}+(x-c)^2+y^2$

$-4a^2+4xc =4a\sqrt{(x-c)^2+y^2}$

$-a^2+xc =a\sqrt{(x-c)^2+y^2}$

$a^4-4axc+x^2c^2=a^2((x-c)^2+y^2)$

$x^2(c^2-a^2)-a^2y^2=a^2(c^2-a^2)$

$\frac{x^2}{a^2}-\frac{y^2}{c^2-a^2}=1(3)$

$\text{Combining formula (1) and (2), we got the total solution}:$

$S(x,y)$

$\frac{x^2}{a^2}-\frac{y^2}{c^2-a^2}=1\text{ for}x>0$

$\text{This solution can be obtained in a shorter way using the definition of hyperbola}$

$\text{let's apply specific data from the task:}$

$M(1.5,0);N(1.5,0)$

$c=0$

$M(0,0);N(0,0)$

$2a= 0.33*4=1.32$

$\text{According to the formula (2)}$

$\sqrt{(x+c)^2+y^2} - \sqrt{(x-c)^2+y^2} =2a$

$\sqrt{(x+0)^2+y^2} - \sqrt{(x-0)^2+y^2} =1.32$

$0=1.32\text{ its false}$

$\text{For the given initial conditions, the problem has no solution}$

Answer:For the given initial conditions, the problem has no solution