Question

Find all the points having an x-coordinate of 4 whose distance from the point (-2,-1) is 10

Answer 1

$A(x_1;\ y_1);\ B(x_2;\ y_2)\\\\the\ distance\ between\ A\ and\ B:\\\\d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\A(-2;-1);\ B(4;\ y);\ d=10\\\\subtitute\\\\\sqrt{(4-(-2))^2+(y-(-1))^2}=10\\\\\sqrt{(4+2)^2+(y+1)^2}=10\\\\\sqrt{6^2+(y+1)^2}=10\\\\\sqrt{36+(y+1)^2}=10\ \ \ |square\ both\ sides\\\\36+(y+1)^2=10^2\\\\36+(y+1)^2=100\ \ \ |subtract\ 36\ from\ both\ sides\\\\(y+1)^2=64\iff y+1=-\sqrt{64}\ or\ y+1=\sqrt{64}\\\\y+1=-8\ or\ y+1=8\ \ \ \ |subtract\ 1\ from\ both\ sides\\\\y=-9\ or\ y=7$

$Answer:{\boxed{(4;-9)\ or\ (4;\ 7)}$