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The straight line \( l_1 \), shown in Figure 1, has equation \( 5y = 4x + 10 \) - Edexcel - A-Level Maths Pure - Question 10 - 2017 - Paper 1
Question 10
The straight line \( l_1 \), shown in Figure 1, has equation \( 5y = 4x + 10 \). The point \( P \) with x coordinate 5 lies on \( l_1 \). The straight line \( l_2 ... show full transcript
Worked Solution & Example Answer:The straight line \( l_1 \), shown in Figure 1, has equation \( 5y = 4x + 10 \) - Edexcel - A-Level Maths Pure - Question 10 - 2017 - Paper 1
Step 1
Find an equation for \( l_2 \), writing your answer in the form \( ax + by + c = 0 \) where \( a, b \) and \( c \) are integers.
Answer
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Identify the gradient of line ( l_1 ): From the equation ( 5y = 4x + 10 ), rewrite it in slope-intercept form, ( y = \frac{4}{5}x + 2 ). Thus, the gradient of line ( l_1 ) is ( \frac{4}{5} ).
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Determine the gradient of line ( l_2 ): Since ( l_2 ) is perpendicular to ( l_1 ), its gradient is the negative reciprocal of ( \frac{4}{5} ), which is ( -\frac{5}{4} ).
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Find the coordinates of point ( P ): Given ( P ) has an x-coordinate of 5, substitute ( x = 5 ) into the equation of ( l_1 ):
[ y = \frac{4}{5} \cdot 5 + 2 = 4 + 2 = 6 ]
Therefore, ( P = (5, 6) ). -
Use point-slope form to find the equation of line ( l_2 ):
The point-slope form is given by:
[ y - y_1 = m(x - x_1) ]
Substitute ( m = -\frac{5}{4} ) and ( (x_1, y_1) = (5, 6) ):
[ y - 6 = -\frac{5}{4}(x - 5) ] -
Rearranging to standard form:
Expanding gives:
[ y - 6 = -\frac{5}{4}x + \frac{25}{4} ]
Multiply through by 4 to eliminate the fraction:
[ 4y - 24 = -5x + 25 ]
Rearranging leads to:
[ 5x + 4y - 49 = 0 ]
Hence, ( a = 5, b = 4, c = -49 ).
Step 2
Calculate the area of triangle \( SPT \).
Answer
To calculate the area of triangle ( SPT ), follow these steps:
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Find the coordinates of points ( S ) and ( T ):
- For point ( S ), set ( y = 0 ) in the equation of line ( l_1 ):
[ 5(0) = 4x + 10 \implies x = -2.5 ]
Therefore, ( S = (-2.5, 0) ). - For point ( T ), set ( y = 0 ) in the equation of line ( l_2 ) (derived above):
[ 5x + 4(0) - 49 = 0 \implies 5x = 49 \implies x = 9.8 ]
Thus, ( T = (9.8, 0) ).
- For point ( S ), set ( y = 0 ) in the equation of line ( l_1 ):
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Calculate the area of triangle ( SPT ) using the formula:
[ \text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height} ]
The base ( ST ) can be computed as the distance between points ( S ) and ( T ) on the x-axis:
[ ST = 9.8 - (-2.5) = 12.3 ] The height is the y-coordinate of point ( P ), which is 6. -
Substituting values into the area formula:
[ \text{Area} = \frac{1}{2} \cdot 12.3 \cdot 6 = 36.9 ]
Thus, the area of triangle ( SPT ) is approximately 36.9 square units.