Question

Please solve it someone
With full explanation please...

Two points are located at A(-4,4) and B(5,3). For two points P, Q on the x-axis and R on the line y=1, find the minimum value of AP+PR+RQ+QB.​

Answer 1

$\large\underline{\text{Solution for Your Question}}$

When R is reflected. (Image 1)

Reflect $R(a,1)$ against $x$-axis. Let the image be R'.

$\implies\overline{PR}=\overline{PR'},\ \overline{RQ}=\overline{RQ'}$

$\implies\overline{AP}+\overline{PR}=\overline{AP}+\overline{PR'}\geq\overline{AR'}\cdots\text{ [1]}$

$\implies\overline{RQ}+\overline{QB}=\overline{R'Q}+\overline{QB}\geq\overline{RB'}\cdots\text{ [2]}$

Adding two inequalities

$\therefore\overline{AP}+\overline{PR}+\overline{RQ}+\overline{QB}\geq\overline{AR'}+\overline{R'B}$

When $\large\overline{AR'}+\overline{R'B}$ has its minimum value. (Image 2)

If all the points are moved to each location

$\implies A'(-4,5),\ B'(5,4),\ R''(a,0)$

$\implies\overline{AR'}+\overline{R'B}\geq\overline{A'R''}+\overline{R''B'}$

Again, if B' is moved to $B''(5,-4)$

$\implies\overline{A'R''}+\overline{R''B''}\geq\overline{A'R''}+\overline{R''B''}\geq\overline{A'B''}$

So, the minimum of $\overline{AP}+\overline{PR}+\overline{RQ}+\overline{QB}$ is equal to $\overline{A'B''}$.

Finding the minimum length. (Image 3)

For the two points $A'(-4,5)$ and $B''(5,-4)$, the length is $9\sqrt{2}$. Hence the minimum length is $9\sqrt{2}$.

Answer 2

Answer:

Solution :

When R is reflected. (Image 1)

Reflect R(a, 1) against x -axis. Let the image

be R'.

PR= PR', RQ = RQ'

AP+ PR = AP + PR'

AR'.

[1]

⇒RQ+QB = R'Q+QB > RB'

[2]

Adding two inequalities

..AP+PR+RQ+QB ≥ AR' + R'B

When AR' + R'B has its minimum value.

(Image 2)

If all the points are moved to each location

Again, if B' is moved to B" (5,-4)

A'(-4,5), B'(5,4), R" (a,0)

AR+R'B≥ A'R" + R"B'

A'R"R"B"

A'R" + R"B" > A'B"

So, the minimum of AP +PR+RQ+QB is equal to A'B" .

Finding the minimum length. (Image 3)

For the two points A'(-4,5) and B" (5,-4),

the length is 9√2. Hence the minimum

length is 9√/2.