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The line p makes an intercept on the x-axis at (a, 0) and on the y-axis at (0, b), where a, b ≠ 0 - Leaving Cert Mathematics - Question 2 - 2019
Question 2
The line p makes an intercept on the x-axis at (a, 0) and on the y-axis at (0, b), where a, b ≠ 0. Show that the equation of p can be written as \( \frac{x}{a} + \fr... show full transcript
Worked Solution & Example Answer:The line p makes an intercept on the x-axis at (a, 0) and on the y-axis at (0, b), where a, b ≠ 0 - Leaving Cert Mathematics - Question 2 - 2019
Step 1
Show that the equation of p can be written as \( \frac{x}{a} + \frac{y}{b} = 1 \)
Answer
To derive the equation of the line p, we start by recognizing that p intercepts the x-axis at (a, 0) and the y-axis at (0, b).
Using the intercept form of a linear equation, we can express the line using the intercepts:
[ \frac{x}{a} + \frac{y}{b} = 1 ]
This represents the desired equation of line p.
Step 2
(i) Write the equation of the line l in terms of m.
Answer
Given that the line l has a slope m and passes through the point A(6, 0), we can use the point-slope form of the equation of a line:
[ y - y_1 = m(x - x_1) ] Substituting the coordinates of point A: [ y - 0 = m(x - 6) ] Therefore, the equation of line l is: [ y = m(x - 6)] Alternatively, we can rewrite it as: [ y = mx - 6m ]
Step 3
(ii) Find the co-ordinates of P in terms of m.
Answer
To find the coordinates of point P where line l cuts line k, we start with the equations we have:
- ( y = m(x - 6) ) (Equation of line l)
- ( 4x + 3y = 25 ) (Equation of line k)
Substituting the expression for y from line l into line k:
[ 4x + 3m(x - 6) = 25 ] Expanding this equation gives: [ 4x + 3mx - 18m = 25 ] Combining like terms leads to: [ (4 + 3m)x = 25 + 18m ] Now solving for x: [ x = \frac{25 + 18m}{4 + 3m} ] To find y, substitute x back into the equation for line l: [ y = m\left( \frac{25 + 18m}{4 + 3m} - 6 \right) ] Simplifying further gives: [ y = \frac{m(25 + 18m - 24 - 18m)}{4 + 3m} = \frac{m(1)}{4 + 3m} = \frac{m}{4 + 3m} ] Thus, the coordinates of P in terms of m are: [ P \left( \frac{25 + 18m}{4 + 3m}, \frac{m}{4 + 3m} \right) ]