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

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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#