Coordinates of point A is (1,3) and B is (5,-3).There is a point P
moving over line y=x+4. Find minimum value of |AP|+|PB|.
I used the distance of a point to a line formula but I found the
answer featuring a square root.Answer must be 6.
Coordinates of point A is (1,3) and B is (5,-3).There is a point P
moving over line y=x+4. Find minimum value of |AP|+|PB|.
I used the distance of a point to a line formula but I found the
answer featuring a square root.Answer must be 6.
Btw, suppose $\sqrt{2x^2+4x+74}+\sqrt{2x^2+2}=k$
Multiply both sides by the conjugate:
$\sqrt{2x^2+4x+74}-\sqrt{2x^2+2}=\dfrac{4x+72}{k}$
Adding these together and squaring both sides gives:
$4(2x^2+4x+74)=\left(k+\dfrac{4x+72}{k}\right)^2$
To minimize $\displaystyle PA+PB$, $\displaystyle P$ must be at the intersection of the line $\displaystyle y=x+4$ with the line $\displaystyle BC$
where $\displaystyle C$ is the symmetric of $\displaystyle A$ with respect to the line $\displaystyle y=x+4$
$\displaystyle C(-1,5)$
$\displaystyle B(5,-3)$
minimum distance = $\displaystyle \sqrt{6^2+8^2}=10$