Photo AI
The line l_1 has equation y = -2x + 3 The line l_2 is perpendicular to l_1 and passes through the point (5, 6) - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 3
Question 7
The line l_1 has equation y = -2x + 3 The line l_2 is perpendicular to l_1 and passes through the point (5, 6). (a) Find an equation for l_2 in the form ax + by + ... show full transcript
Worked Solution & Example Answer:The line l_1 has equation y = -2x + 3 The line l_2 is perpendicular to l_1 and passes through the point (5, 6) - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 3
Step 1
Find an equation for l_2 in the form ax + by + c = 0, where a, b and c are integers.
Answer
To find the equation of line l_2, we first need to determine the slope of l_1. The equation of l_1 is in slope-intercept form (y = mx + b) where the slope m = -2. The slope of a line perpendicular to l_1 will be the negative reciprocal of -2, which is ( \frac{1}{2} ).
Using the point-slope form of the equation for l_2, we have:
[ y - 6 = \frac{1}{2}(x - 5) ]
Expanding this:
[ y - 6 = \frac{1}{2}x - \frac{5}{2} ]
Now, rearranging it into the required form:
[ 2y - 12 = x - 5 ]
[ x - 2y + 7 = 0 ]
Thus, the equation for l_2 is ( x - 2y + 7 = 0 ).
Step 2
Find the x-coordinate of A and the y-coordinate of B.
Answer
To find the coordinates of point A where the line l_1 crosses the x-axis, set y = 0 in the equation of l_1:
[ 0 = -2x + 3 ]
Solving for x gives:
[ 2x = 3 \quad \Rightarrow \quad x = \frac{3}{2} ]
Thus, point A is ( \left( \frac{3}{2}, 0 \right) ).
Next, to find the y-coordinate of point B where l_1 crosses the y-axis, set x = 0:
[ y = -2(0) + 3 = 3 ]
So, point B is ( (0, 3) ).
Step 3
find the area of the triangle OAB.
Answer
The area of triangle OAB can be calculated using the formula:
[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ]
In triangle OAB, O is (0,0), A is ( \left( \frac{3}{2}, 0 \right) ) and B is (0, 3). The base OA has length ( \frac{3}{2} ) and the height OB has length 3.
Thus:
[ \text{Area} = \frac{1}{2} \times \frac{3}{2} \times 3 = \frac{9}{4} \text{ units}^2 ]