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

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2 Answers
Jan 29, 2018

a=4/5a=45

Explanation:

"find the coordinates of P and Q"find the coordinates of P and Q

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

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

x=0toy/2=1rArry=2larrcolor(red)"y-intercept"x=0y2=1y=2y-intercept

y=0tox/3=1rArrx=3larrcolor(red)"x-intercept"y=0x3=1x=3x-intercept

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

(a)(a)

m_(QR)=1/2" and "R=(2a,y)mQR=12 and R=(2a,y)

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

•color(white)(x)m=(y_2-y_1)/(x_2-x_1)xm=y2y1x2x1

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

rArr(y-2)/(2a-0)=(y-2)/(2a)=1/2y22a0=y22a=12

rArr2(y-2)=2a2(y2)=2a

rArry-2=arArry=a+2y2=ay=a+2

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

(b)(b)

"using the gradient formula with"using the gradient formula with

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

rArr(a+2)/(2a-3)=-2a+22a3=2

rArra+2=-4a+6a+2=4a+6

rArr5a=4rArra=4/55a=4a=45

Jan 29, 2018

See below.

Explanation:

First the coordinates for PP and QQ

x=0 rArr y/2+x/3=1 rArr y = 2 rArr P = (x_P,y_P)=(0,2)x=0y2+x3=1y=2P=(xP,yP)=(0,2)
y=0 rArr y/2+x/3=1 rArr x = 3 rArr Q = (x_Q,y_Q)=(3,0)y=0y2+x3=1x=3Q=(xQ,yQ)=(3,0)

Now calling R = (x_R,y_R)R=(xR,yR)

for [QR][QR]

(y_R-y_Q)/(x_R-x_Q) = 1/2yRyQxRxQ=12 or

(y_R-0)/(2a-3)=1/2 rArr y_R = a-3/2yR02a3=12yR=a32

and for [PR][PR]

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

(y_R-2)/(2a-0) = -2 rArr y_R = 2-4ayR22a0=2yR=24a

but

a-3/2 = 2-4a rArr a = 7/10a32=24aa=710