Question

What is the value of a if the gradient of PR is -2?

Answer 1

`a=4/5`

Explanation:

`"find the coordinates of P and Q"`

`• " let x = 0, in equation for y-intercept"`

`• " let y = 0, in equation for x-intercept"`

`x=0toy/2=1rArry=2larrcolor(red)"y-intercept"`

`y=0tox/3=1rArrx=3larrcolor(red)"x-intercept"`

`rArrP=(3,0)" and "Q=(0,2)`

`(a)`

`m_(QR)=1/2" and "R=(2a,y)`

`"using the "color(blue)"gradient formula"`

`•color(white)(x)m=(y_2-y_1)/(x_2-x_1)`

`"with "Q=(0,2)" and "R=(2a,y)`

`rArr(y-2)/(2a-0)=(y-2)/(2a)=1/2`

`rArr2(y-2)=2a`

`rArry-2=arArry=a+2`

`rArrR=(2a,a+2)`

`(b)`

`"using the gradient formula with"`

`P(3,0)" and "R(2a,a+2)`

`rArr(a+2)/(2a-3)=-2`

`rArra+2=-4a+6`

`rArr5a=4rArra=4/5`

Answer 2

See below.

Explanation:

First the coordinates for `P` and `Q`

`x=0 rArr y/2+x/3=1 rArr y = 2 rArr P = (x_P,y_P)=(0,2)`
`y=0 rArr y/2+x/3=1 rArr x = 3 rArr Q = (x_Q,y_Q)=(3,0)`

Now calling `R = (x_R,y_R)`

for `[QR]`

`(y_R-y_Q)/(x_R-x_Q) = 1/2` or

`(y_R-0)/(2a-3)=1/2 rArr y_R = a-3/2`

and for `[PR]`

`(y_R-y_P)/(x_R-x_P) = -2` or

`(y_R-2)/(2a-0) = -2 rArr y_R = 2-4a`

but

`a-3/2 = 2-4a rArr a = 7/10`