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

enter image source here

2 Answers
Jan 29, 2018

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

Jan 29, 2018

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