Photo AI
The line L has equation 3y - 4x = 21 The point P has coordinates (15, 2) Find the equation of the line perpendicular to L which passes through P - AQA - A-Level Maths Mechanics - Question 5 - 2021 - Paper 1
Question 5
The line L has equation 3y - 4x = 21 The point P has coordinates (15, 2) Find the equation of the line perpendicular to L which passes through P. [2 marks] (b) He... show full transcript
Worked Solution & Example Answer:The line L has equation 3y - 4x = 21 The point P has coordinates (15, 2) Find the equation of the line perpendicular to L which passes through P - AQA - A-Level Maths Mechanics - Question 5 - 2021 - Paper 1
Step 1
Find the equation of the line perpendicular to L which passes through P.
Answer
To start, we need to rewrite the equation of line L in slope-intercept form.
-
Convert the equation: 3y - 4x = 21
- Rearranging gives us: 3y = 4x + 21
- Dividing by 3: y = \frac{4}{3}x + 7
-
The slope of line L is \frac{4}{3}. The negative reciprocal (perpendicular slope) is -\frac{3}{4}.
-
Using point P(15, 2), we can use the point-slope form of the equation to find the equation of the perpendicular line:
- y - y_1 = m(x - x_1)
- y - 2 = -\frac{3}{4}(x - 15)
-
Simplifying this gives:
- y - 2 = -\frac{3}{4}x + \frac{45}{4}
- y = -\frac{3}{4}x + \frac{45}{4} + 2
- y = -\frac{3}{4}x + \frac{45}{4} + \frac{8}{4}
- y = -\frac{3}{4}x + \frac{53}{4}.
Thus, the equation of the line perpendicular to L and passing through P is: y = -\frac{3}{4}x + \frac{53}{4}.
Step 2
Hence, find the shortest distance from P to L.
Answer
To find the shortest distance from point P(15, 2) to line L, we can use the formula for the distance from a point to a line given by Ax + By + C = 0:
The line equation can be rearranged as: (4x - 3y + 21 = 0) with A = 4, B = -3, and C = 21.
We apply the distance formula: [ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} ] where ( (x_0, y_0) ) is the point P.
Substituting in the values: [ d = \frac{|4(15) - 3(2) + 21|}{\sqrt{4^2 + (-3)^2}} ] [ = \frac{|60 - 6 + 21|}{\sqrt{16 + 9}} ] [ = \frac{|75|}{5} = 15. ]
Therefore, the shortest distance from P to L is 15.