The line through (2, 5) and (–3, –2) is perpendicular to the line through (4, –1) and (x,3).The value of ‘x’ is
Answers
EXPLANATION.
The line through (2,5) and (-3,-2).
Perpendicular to the line through (4,-1) and (x,3).
As we know that,
Slope of the line = (y₂ - y₁)/(x₂ - x₁) = m.
Slope of the line (2,5) and (-3,-2) = m₁.
⇒ (-2 - 5)/(-3 - 2) = -7/-5 = 7/5.
Slope of the line (4,-1) and (x, 3). = m₂
⇒ (3 - (-1))/(x - 4) = (3 + 1)/(x - 4) = 4/(x - 4).
Two lines are perpendicular, if m₁ m₂ = - 1.
⇒ (7/5) x (4/x - 4) = - 1.
⇒ 28/(5x - 20) = - 1.
⇒ 28 = - 1(5x - 20).
⇒ 28 = - 5x + 20.
⇒ 28 - 20 = - 5x.
⇒ 8 = - 5x.
⇒ x = - 8/5.
MORE INFORMATION.
The angle between two straight lines.
(1) = tanθ = |m₁ - m₂/1 + m₁ m₂|.
(2) = Two lines are parallel if, m₁ = m₂.
(3) = Two lines are perpendicular if, m₁ m₂ = - 1.
(4) = Two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are coincident only and only if, a₁/a₂ = b₁/b₂ = c₁/c₂.
Given :-
The line through (2, 5) and (–3, –2) is perpendicular to the line through (4, –1) and (x,3).
To Find :-
Value of x
Solution :-
Since they are perpendicular to each other. So, their product will be -1
For S₁
Slope = y₂ - y₁/x₂ - x₁
Slope = [(-2) - (5)]/[(-3) + (-2)]
Slope = [(-2 - 5)]/[(-3 - 2)]
Slope = [-7]/[-5]
Slope = 7/5
For S₂
Slope = [3 - (-1)]/[x - 4]
Slope = 3 + 1/x - 4
Slope = 4/x - 4
Now
7/5 × 4/x - 4 = -1
7 × 4/5(x - 4) = -1
28/5x - 20 = -1
28 = -1(5x - 20)
28 = -5x + 20
28 - 20 = -5x
8 = -5x
8/-5 = x
-8/5 = x