Question-20744

Question Number 20744 by mondodotto@gmail.com last updated on 02/Sep/17

Answered by $@ty@m last updated on 02/Sep/17

ATQ, Let the equation of the st. line be y=mx+c −−(1) It passes through (4,−2) ∴ −2=4m+c ⇒c=−2−4m ∴ the equation of the st. line is y=mx−4m−2 −−(2) Put y=0 ⇒x=((4m+2)/m) ∴ coordinates of R=(((4m+2)/m),0) Put x=0 ⇒y=−4m−2 ∴ coordinates of S=(0, −4m−2) T is mid point of RS ∴ coordinates of T=(((1+2m)/2), −2m−1)=(p,q), say ⇒p=((1+2m)/2), q=−2m−1 ⇒p=((−q)/2) ⇒2p+q=0 ∴locus of T(p,q) is 2x+y=0 which satisfies (0,0) ⇒ it passes through origin.

$${ATQ}, \\ $$$${Let}\:{the}\:{equation}\:{of}\:{the}\:{st}.\:{line}\:{be} \\ $$$${y}={mx}+{c}\:−−\left(\mathrm{1}\right) \\ $$$${It}\:{passes}\:{through}\:\left(\mathrm{4},−\mathrm{2}\right) \\ $$$$\therefore\:−\mathrm{2}=\mathrm{4}{m}+{c} \\ $$$$\Rightarrow{c}=−\mathrm{2}−\mathrm{4}{m} \\ $$$$\therefore\:{the}\:{equation}\:{of}\:{the}\:{st}.\:{line}\:{is} \\ $$$${y}={mx}−\mathrm{4}{m}−\mathrm{2}\:−−\left(\mathrm{2}\right) \\ $$$${Put}\:{y}=\mathrm{0} \\ $$$$\Rightarrow{x}=\frac{\mathrm{4}{m}+\mathrm{2}}{{m}} \\ $$$$\therefore\:{coordinates}\:{of}\:{R}=\left(\frac{\mathrm{4}{m}+\mathrm{2}}{{m}},\mathrm{0}\right) \\ $$$${Put}\:{x}=\mathrm{0} \\ $$$$\Rightarrow{y}=−\mathrm{4}{m}−\mathrm{2} \\ $$$$\therefore\:{coordinates}\:{of}\:{S}=\left(\mathrm{0},\:−\mathrm{4}{m}−\mathrm{2}\right) \\ $$$${T}\:{is}\:{mid}\:{point}\:{of}\:{RS} \\ $$$$\therefore\:{coordinates}\:{of}\:{T}=\left(\frac{\mathrm{1}+\mathrm{2}{m}}{\mathrm{2}},\:−\mathrm{2}{m}−\mathrm{1}\right)=\left({p},{q}\right),\:{say} \\ $$$$\Rightarrow{p}=\frac{\mathrm{1}+\mathrm{2}{m}}{\mathrm{2}},\:{q}=−\mathrm{2}{m}−\mathrm{1} \\ $$$$\Rightarrow{p}=\frac{−{q}}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{2}{p}+{q}=\mathrm{0} \\ $$$$\therefore{locus}\:{of}\:{T}\left({p},{q}\right)\:{is} \\ $$$$\mathrm{2}{x}+{y}=\mathrm{0} \\ $$$${which}\:{satisfies}\:\left(\mathrm{0},\mathrm{0}\right) \\ $$$$\Rightarrow\:{it}\:{passes}\:{through}\:{origin}. \\ $$

Commented by mondodotto@gmail.com last updated on 02/Sep/17