Q.61 - Calculus

Question

Use the arc length formula for parametric equations and your answer to exercise $55$to find the distance between the points $(1,-3)$and $(6,7)$ Verify your answer by using the distance formula to compute the distance.

Step-by-Step Solution

Answer

Therefore, the distance between the points is $5\sqrt{5}$

Step:1 Given information

The points from$(1,-3)$to $(6,7).$

Step 2: Calculation

The goal is to determine the parametric equations for the line segment that connects the two points. Then, using the distance formula, calculate the distance between the spots .The formula for the line segment joining the pair of points $(a, b)$to $(c, d)$is as follows, $x=a+(c-a)t,y=b+(d-b)t,t\in [0,1].$

Here, to find the parametric equations,, substitute the given values in the equation of line segment.

Now take the points $(1,-3),(6,7)$.

Substituting the values in the equation, $x=a+(c-a) t$we get,

$x=1+(6-1) t$

$x=1+5 t$Since $a=1, b=-3, c=6, d=7$now take the points $(1,-3),(6,7)$.

Substitute the values in the equation $y=b+(d-b) t$, we get

$$\begin{gathered} y=-3+(7-(-3\left)\right) t \\ y=-3+(7+3) t \\ y=-3+10 t \end{gathered}$$

Step:3 Further calculation

Thus. the parametric equations for the line segment joining the pair of points $(1,-3)(6,7)$are$x=1+5t,y=-3+10r,t\in [0,1]$.

Therefore, the parametric equations are$x=1+5t,y=-3+10t,t\in [0,1]$.

Now we need to find the distance between the points $(1,-3),(6,7)$.

The formula for the distance $=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Then the distance$=\sqrt{(6-1{)}^2+(7-(-3){)}^2}$ since $\left.x_1=1,x_2=6,y_1=-3,y_2=7\right]$

$$\begin{gathered} =\sqrt{5^2+{10}^2} \\ =\sqrt{125} \end{gathered}$$