2. Find the equation of the line which passes through [tex]\(A(-1,4)\)[/tex] and is perpendicular to the line which passes through [tex]\(A\)[/tex] and [tex]\(B(5, 2)\)[/tex].

3. Find the distance of the point [tex]\(P(3,4)\)[/tex] from the line [tex]\(y = -2x + 3\)[/tex].

4. Two ships follow straight line courses at steady speeds. The position of Ship 1 is [tex]\((3t - 10, t + 4)\)[/tex] and the position of Ship 2 is [tex]\((2t - 3, -t + 13)\)[/tex], where [tex]\(t\)[/tex] represents time (in hours) and distances are in km. What is the closest distance that the ships approach each other and at what time does this occur?

5. Consider the two circles:
[tex]\[
(x − 3)^2 + (y + 4)^2 = 53
\][/tex]
[tex]\[
(x + 2)^2 + (y - 1)^2 = 13
\][/tex]
Determine their intersection points and a line joining them.



Answer :

Let's solve each of the problems step by step:

### 2. Equation of the Perpendicular Line
1. Find the slope of the line through points A(-1, 4) and B(5, 2):
- The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is [tex]\( m = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex].
- Here, [tex]\( m = \frac{2 - 4}{5 + 1} = \frac{-2}{6} = -\frac{1}{3} \)[/tex].

2. Find the slope of the line perpendicular to it:
- The slope of a line perpendicular to a line with slope [tex]\( m \)[/tex] is the negative reciprocal of [tex]\( m \)[/tex].
- Thus, the perpendicular slope is [tex]\( m_{\perp} = -\frac{1}{-\frac{1}{3}} = 3 \)[/tex].

3. Use the point-slope form for the perpendicular line:
- The point-slope form is [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, using point [tex]\( A(-1, 4) \)[/tex] and the perpendicular slope (3), the equation is [tex]\( y - 4 = 3(x + 1) \)[/tex].
- Expanding this: [tex]\( y - 4 = 3x + 3 \)[/tex] which simplifies to [tex]\( y = 3x + 7 \)[/tex].

Final Equation: [tex]\( y = 3x + 7 \)[/tex]

### 3. Distance from Point P(3,4) to Line y = -2x + 3
1. Use the distance formula from a point to a line [tex]\(Ax + By + C = 0\)[/tex]:
- For the line [tex]\( y = -2x + 3 \)[/tex], rewrite it as [tex]\( 2x + y - 3 = 0 \)[/tex].
- [tex]\( A = 2 \)[/tex], [tex]\( B = 1 \)[/tex], [tex]\( C = -3 \)[/tex].

2. Plug the values into the distance formula:
- Distance [tex]\( D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \)[/tex].
- For point [tex]\( P(3,4) \)[/tex], distance [tex]\( D = \frac{|2(3) + 1(4) - 3|}{\sqrt{2^2 + 1^2}} = \frac{|6 + 4 - 3|}{\sqrt{5}} = \frac{7}{\sqrt{5}} = 2.23606797749979 \)[/tex].

Final Distance: [tex]\( 2.23606797749979 \)[/tex]

### 4. Closest Distance and Time Between Two Ships
1. Express the positions of Ship 1 and Ship 2 as functions of time [tex]\( t \)[/tex]:
- Ship 1: [tex]\((3t - 10, t + 4)\)[/tex]
- Ship 2: [tex]\((2t - 3, -t + 13)\)[/tex]

2. Define the distance function [tex]\( D(t) \)[/tex]:
- [tex]\( D(t) = \sqrt{((2t - 3) - (3t - 10))^2 + ((-t + 13) - (t + 4))^2} \)[/tex].
- Simplified: [tex]\( D(t) = \sqrt{(-t + 7)^2 + (-2t + 9)^2} \)[/tex].

3. Minimize [tex]\( D(t) \)[/tex]:
- By using minimization techniques, the closest distance is obtained.

Closest Distance: [tex]\( 2.236067977500005 \)[/tex], at time [tex]\( t \approx 4.99999956092568 \)[/tex] hours

### 5. Intersection Points of Two Circles
1. Equations of the circles:
- Circle 1: [tex]\( (x - 3)^2 + (y + 4)^2 = 53 \)[/tex]
- Circle 2: [tex]\( (x + 2)^2 + (y - 1)^2 = 13 \)[/tex]

2. Solve the system of equations:
- Solving these simultaneously, we find the intersection points.

Intersection Points: [tex]\( (-4, -2) \)[/tex] and [tex]\( (1, 3) \)[/tex]

3. Equation of the line passing through these points:
- Slope: [tex]\( m = \frac{3 - (-2)}{1 - (-4)} = \frac{5}{5} = 1 \)[/tex]
- Using point-slope form with point [tex]\( (-4, -2) \)[/tex]:
[tex]\( y - (-2) = 1(x + 4) \)[/tex]
[tex]\( y + 2 = x + 4 \)[/tex]
[tex]\( y = x + 2 \)[/tex]

Final Line Equation: [tex]\( y = x + 2 \)[/tex]