Question

Find the point on the line 5x + y = 7 that is closest to the point (−3, 2).

Answer 1

Answer:

$(\dfrac{11}{13},\dfrac{36}{13})$.

Step-by-step explanation:

The given equation of line is

$5x+y=7$         .... (1)

If a line is defined as ax+by=c, then the slope of the line is -a/b.

We need to find the point on the line 5x + y = 7 that is closest to the point (−3, 2).

Draw a perpendicular line from the point (-3,2) on the line 5x + y = 7.

For the given line a=5 and b=1. So, the slope of given line is -5.

Product of slopes of two perpendicular lines is -1. Slope of given line -5, it means the slope of perpendicular line is 1/5.

The slope of perpendicular line is  1/5 and it passes through the point (-3,2). So, the equation of line is

$y-y_1=m(x-x_1)$

where, m is slope.

$y-2=\frac{1}{5}(x-(-3))$

Multiply both sides by 5.

$5y-10=x+3$

$5y-x=10+3$

$5y-x=13$     .  .... (2)

The intersection point of line (1) and (2) is the point on the line 5x + y = 7 that is closest to the point (−3, 2).

On solving (1) and (2) we get

$x=\dfrac{11}{13}, y=\dfrac{36}{13}$

Therefore, the point $(\dfrac{11}{13},\dfrac{36}{13})$ on the line 5x + y = 7 that is closest to the point (−3, 2).

Answer 2

Following are the calculation to the points:

Given:

Line: $5x + y = 7$

point: $(-3, 2)$

To find:

point=?

Solution:

Assuming the point $P(h,k)$ is closest to the distance from $P(h,k)\ to\ (-3,2)$

$D= \sqrt{(h+3)^2+(k-2)^2}$                   using the distance formula

Equation of line:

$\to 5h +k= 7 \\\\\to k= 7-5h$

Substituting the value of $k=7-5h \ in\ D= \sqrt{(h+3)^2+(k-2)^2}$

$D= \sqrt{(h+3)^2+(7-5h-2)^2}$

   $= \sqrt{(h+3)^2+(-5h+5)^2} \\\\= \sqrt{h^2+9+6h +25h^2 +25 -50h} \\\\= \sqrt{26h^2+34-44h}$

Differentitate D with respect to h

$\frac{dD}{dh} =\frac{44h-34}{2 \sqrt{26h^2-44h +34}}$

Set $\ \frac{dD}{dh}=0$

Then,

$44h-34= 0\\\\44h=34\\\\\to h=\frac{34}{44}\=\frac{17}{22}$

So, $k = 7-5h$

         $= 7-5 \frac{17}{22}\\\\= 7-\frac{85}{22}\\\\= \frac{154-85}{22}\\\\= \frac{69}{22}$

Therefore, the points are "$(\frac{69}{22}, \frac{17}{22} )$"

Learn more:

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