Find the co-ordinates of the foot of the perpendicular
drawn from the point (2, 3) on the line 3x + 4y + 8 = 0
Answers
EXPLANATION.
Co-ordinates of the foot of the perpendicular drawn from the point = (2,3).
on the line = 3x + 4y + 8 = 0.
As we know that,
Slope of the perpendicular line = b/a.
Slope of line : 3x + 4y + 8 = 0 is 4/3.
Slope = 4/3.
Equation of line,
⇒ (y - y₁) = m(x - x₁).
Put the values in the equation, we get.
⇒ (y - 3) = 4/3(x - 2).
⇒ 3(y - 3) = 4(x - 2).
⇒ 3y - 9 = 4x - 8.
⇒ 3y - 4x - 9 + 8 = 0.
⇒ 3y - 4x - 1 = 0.
As we know that,
⇒ 3x + 4y + 8 = 0. ⇒ (1).
⇒ 3y - 4x - 1 = 0. ⇒ (2).
Solving equation (1) & (2), we get.
Multiply equation (1) by 4 & (2) by 3, we get.
⇒ 12x + 16y + 32 = 0.
⇒ 9y - 12x - 3 = 0.
⇒ 25y + 29 = 0.
⇒ y = -29/25.
Put the value of y = -29/25 in equation (1), we get.
⇒ 3x + 4y + 8 = 0.
⇒ 3x + 4(-29/25) + 8 = 0.
⇒ 3x - 116/25 + 8 = 0.
⇒ 75x - 116 + 200 = 0.
⇒ 75x + 84 = 0.
⇒ 75 = -84.
⇒ x = -28/25.
Their Co-ordinates = (-28/25, -29/25).
✬ QUESTION.
- ❍ Co-ordinates of the foot of the perpendicular drawn from the point = (2,3)on the line = 3x + 4y + 8 = 0.
✬ FORMULA USED.
- ❍ Slope of the perpendicular line = b/a.
✬ SOLUTION.
- ❍ Slope of line : 3x + 4y + 8 = 0 is 4/3.
☯ Equation of line,
➯ (y - y₁) = m(x - x₁)
☯ Substitute the values,
➯ (y - 3) = 4/3(x - 2)
➯ 3(y - 3) = 4(x - 2)
➯ 3y - 9 = 4x - 8
➯ 3y - 4x - 9 + 8 = 0
➯ 3y - 4x - 1 = 0
☯ Then,
➯ 3x + 4y + 8 = 0 ..(1)
➯ 3y - 4x - 1 = 0 ..(2)
☯ Solve the equation (1) & (2),
☯ Then, multiply equation (1) by 4 & (2) by 3,
➯ 12x + 16y + 32 = 0
➯ 9y - 12x - 3 = 0
➯ 25y + 29 = 0
➯ y => -29/25
☯ Then, put the value of y = -29/25 in equation (1),
➯ 3x + 4y + 8 = 0
➯ 3x + 4(-29/25) + 8 = 0
➯ 3x - 116/25 + 8 = 0
➯ 75x - 116 + 200 = 0
➯ 75x + 84 = 0
➯ 75 = -84
➯ x = -28/25
❒ Hence, The Co-ordinates are ➯ (-28/25, -29/25)