Coordinate Geometry of St.Line

1.
Given that A(2,5) and B(-3,-5) are two points on the rectangularcoordinate plane.
Find the angle between AB and the x-axis, correct the answerto the nearest 0.1.
Given that a straight line passing through D(7,1) and E(-3,k) is perpendicular to AB. Find the value of k.

Denote θ as the angle between ABand the x-axis.

Slope of AB
= (5 + 5)/(2 + 3)
= 2

tanθ = 2
θ = 63.4°

DE⊥AB, then slope of DE :
(1 - k)/(7 + 3) = -1/2
2 - 2k = -10
2k = 12
k = 6


2.
A straight line L passes through (3,1) and (-1,7). Given that L cuts thex-axis at P and the y-axis at Q.
(a) Find the coordinates of P and Q.
(b) Find the area of Triangle OPQ where O is the origin.

(a)
Let (a, 0) and (0, b) be the coordinates of A and B respectively.

Slope of L :
(1 - 0)/(3 - a) = (1 - 7)/(3 + 1)
1/(3 - a) = -3/2
-9 + 3a = 2
a = 11/3

Slope of L :
(1 - b)/(3 - 0) = (1 - 7)/(3 + 1)
(1 - b)/3 = -3/2
2 - 2b = -9
b = 11/2

The coordinates of P = (11/3, 0)
The coordinates of Q = (0, 11/2)

(b)
Area of ΔOPQ
= (1/2) * (11/3) * (11/2)
= 121/12(aq. units)

Thank you So much

Given that A(2,5) and B(-3,-5) are two points on the rectangular coordinate plane.
Find the angle between AB and the X-axis, correct the answer to the nearest 0.1.
Let the angle be x.
tanx=(-5-5)/(-3-2)
x=63.4°(corr.to nearest 0.1)
Given that a straight line passing through D(7,1) and E(-3,K) is perpendicular toAB. Find the value of K.
slope of AB=(-5-5)/(-3-2)=2
∴(K-1)/(-3-7)x2=-1
K=6

A straight line L passes through (3,1) and (-1,7). Given that L cuts the x-axis at P and the y-axis at Q.
(a) Find the coordinates of P and Q.
(b) Find the area of Triangle OPQ where O is the origin.
(a) Let P(x, 0) be the coordinates of P.
(7-1)/(-1-3)=(0-1)/(x-3)
x=11/3
∴P(x, 0)=(11/3, 0)
Let Q(0, y) be the coordinates of Q.
(7-1)/(-1-3)=(y-1)/(0-3)
y=11/2
∴P(0, y)=(0, 11/2)
(b) The area of Triangle OPQ=(11/3)x(11/2)/2=121/12 square units