Question and Answers Forum

All Questions      Topic List

Matrices and Determinants Questions

Previous in All Question      Next in All Question      

Previous in Matrices and Determinants      Next in Matrices and Determinants      

Question Number 95260 by i jagooll last updated on 24/May/20

if the line 3x+2y−1=0 transformed  by matrix A= (((1   a)),((b   2)) ) such that  the image is the line 2x+8y+c=0  find the value of a×b×c

$$\mathrm{if}\:\mathrm{the}\:\mathrm{line}\:\mathrm{3x}+\mathrm{2y}−\mathrm{1}=\mathrm{0}\:\mathrm{transformed} \\ $$$$\mathrm{by}\:\mathrm{matrix}\:\mathrm{A}=\begin{pmatrix}{\mathrm{1}\:\:\:\mathrm{a}}\\{\mathrm{b}\:\:\:\mathrm{2}}\end{pmatrix}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{image}\:\mathrm{is}\:\mathrm{the}\:\mathrm{line}\:\mathrm{2x}+\mathrm{8y}+\mathrm{c}=\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}×\mathrm{b}×\mathrm{c}\: \\ $$

Commented by mr W last updated on 24/May/20

(x,y)→(u,v) with  A= (((1   a)),((b   2)) )   u=x+ay  v=bx+2y  2(x+ay)+8(bx+2y)+c=0  2(1+4b)x+2(a+8)y+c=0  ≡3x+2y−1=0  2(1+4b)=3 ⇒b=(1/8)  2(a+8)=2 ⇒a=−7  c=−1  ⇒a×b×c=(7/8)

$$\left({x},{y}\right)\rightarrow\left({u},{v}\right)\:{with}\:\:\mathrm{A}=\begin{pmatrix}{\mathrm{1}\:\:\:\mathrm{a}}\\{\mathrm{b}\:\:\:\mathrm{2}}\end{pmatrix}\: \\ $$$${u}={x}+{ay} \\ $$$${v}={bx}+\mathrm{2}{y} \\ $$$$\mathrm{2}\left({x}+{ay}\right)+\mathrm{8}\left({bx}+\mathrm{2}{y}\right)+{c}=\mathrm{0} \\ $$$$\mathrm{2}\left(\mathrm{1}+\mathrm{4}{b}\right){x}+\mathrm{2}\left({a}+\mathrm{8}\right){y}+{c}=\mathrm{0} \\ $$$$\equiv\mathrm{3}{x}+\mathrm{2}{y}−\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{2}\left(\mathrm{1}+\mathrm{4}{b}\right)=\mathrm{3}\:\Rightarrow{b}=\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\mathrm{2}\left({a}+\mathrm{8}\right)=\mathrm{2}\:\Rightarrow{a}=−\mathrm{7} \\ $$$${c}=−\mathrm{1} \\ $$$$\Rightarrow{a}×{b}×{c}=\frac{\mathrm{7}}{\mathrm{8}} \\ $$

Commented by mr W last updated on 24/May/20

i am not sure if i understand the  question correctly. please check!

$${i}\:{am}\:{not}\:{sure}\:{if}\:{i}\:{understand}\:{the} \\ $$$${question}\:{correctly}.\:{please}\:{check}! \\ $$

Commented by i jagooll last updated on 24/May/20

thank you sir.   you and sir john got the same result

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}.\: \\ $$$$\mathrm{you}\:\mathrm{and}\:\mathrm{sir}\:\mathrm{john}\:\mathrm{got}\:\mathrm{the}\:\mathrm{same}\:\mathrm{result} \\ $$

Answered by john santu last updated on 24/May/20

 (((x′)),(y^′ ) ) =  (((1   a)),((b   2)) )  ((x),(y) ) =  (((x+ay)),((bx+2y)) )  image of the line 2x′+8y′+c = 0  ⇒ 2(x+ay)+8(bx+2y)+c = 0  it similar to line 3x+2y−1 = 0  ⇒(2+8b)x+(2a+16)y+c = 0  we get 2+8b = 3 ⇒b = (1/8)  2a+16 = 2 ⇒ a = −7   and c = −1 so a×b×c = (7/8) .

$$\begin{pmatrix}{\mathrm{x}'}\\{\mathrm{y}^{'} }\end{pmatrix}\:=\:\begin{pmatrix}{\mathrm{1}\:\:\:\mathrm{a}}\\{\mathrm{b}\:\:\:\mathrm{2}}\end{pmatrix}\:\begin{pmatrix}{\mathrm{x}}\\{\mathrm{y}}\end{pmatrix}\:=\:\begin{pmatrix}{\mathrm{x}+\mathrm{ay}}\\{\mathrm{bx}+\mathrm{2y}}\end{pmatrix} \\ $$$$\mathrm{image}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{2x}'+\mathrm{8y}'+\mathrm{c}\:=\:\mathrm{0} \\ $$$$\Rightarrow\:\mathrm{2}\left(\mathrm{x}+\mathrm{ay}\right)+\mathrm{8}\left(\mathrm{bx}+\mathrm{2y}\right)+\mathrm{c}\:=\:\mathrm{0} \\ $$$$\mathrm{it}\:\mathrm{similar}\:\mathrm{to}\:\mathrm{line}\:\mathrm{3x}+\mathrm{2y}−\mathrm{1}\:=\:\mathrm{0} \\ $$$$\Rightarrow\left(\mathrm{2}+\mathrm{8b}\right)\mathrm{x}+\left(\mathrm{2a}+\mathrm{16}\right)\mathrm{y}+\mathrm{c}\:=\:\mathrm{0} \\ $$$$\mathrm{we}\:\mathrm{get}\:\mathrm{2}+\mathrm{8b}\:=\:\mathrm{3}\:\Rightarrow\mathrm{b}\:=\:\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\mathrm{2a}+\mathrm{16}\:=\:\mathrm{2}\:\Rightarrow\:\mathrm{a}\:=\:−\mathrm{7}\: \\ $$$$\mathrm{and}\:\mathrm{c}\:=\:−\mathrm{1}\:\mathrm{so}\:\mathrm{a}×\mathrm{b}×\mathrm{c}\:=\:\frac{\mathrm{7}}{\mathrm{8}}\:.\: \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com