Question

If line L in the xy-coordinate plane has a positive slope, what is the x-intercept of L ? (1) There are different points (a, b) and (c, d) on line L such that ad = bc. (2) There are constants m and n such that the points (m, n) and (–m, –n) are both on line L.

Answer 1

The x-intercept of L is 0

Explanation:

1) If (a,b) and (c,d) belongs to theline L, then the equation of the line using two-points formula will be:

`y-b=((b-d)/(a-c) ).(x-a)`

if we want to find the x-intercept of L we should set y=0.

`0-b=((b-d)/(a-c) ).(x-a)`

getting x from that equation we will have :

`x=-b.(a-c)/(b-d)+a`

using distributive propertie and common denominator will be obtain

x=
`x= (bc-ad)/(b-d)`

as we know that ad=bc the numerator will be equal to zero. Then x=0.

2) Using the same equation of line but using the points(m,n) and (-m, -n) we will set it as:

`y-n=(-2n)/(-2m).(x-m)`

if we want to find the x-intercept of L we should set y=0.

`0-n=(-2n)/(-2m).(x-m)`

getting x from that equation we will have :

`-n=(n)/(m)(x-m)`

`x= -n.(m)/(n) + m`

them

x = 0