Question

If A and B are (-2,-2) and (2,-4) respectively,find the coordinates of P such that AP = 3/7 AB and p P lies on the line segment AB.

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Answer 1

Question :-

If A and B are (-2,-2) and (2,-4) respectively,find the coordinates of P such that AP = 3/7 AB and p P lies on the line segment AB.

Given that :-

• A and B are (-2,-2) and (2,-4)
• ${ \rm{AP = \frac{3}{7} \: AB}}$

To Find :-

• Coordinates of P ?

Solution :-

• Let the co - ordinates of P be (x,y)

$\dashrightarrow{ \rm{AP = \frac{3}{7} \: AB - - - - (1)Eq.}}$

$\dashrightarrow{ \rm{AB = AP + PB }}$

$\dashrightarrow{ \rm{AB = \frac{3}{7} AB + PB }}$

$\dashrightarrow{ \rm{PB = AB - \frac{3}{7} \: AB }}$

$\dashrightarrow{ \rm{PB = \frac{7 AB - 3 AB}{7} }}$

$\dashrightarrow{ \rm{PB = \frac{4}{7} AB - - - - (2)Eq.}}$

${ \rm{Divide \: (1) \: and \: (2) \: Eq.}}$

$\dashrightarrow { \rm{\frac{AP}{PB} = \frac{ \frac{3}{ \cancel7} \: \: \cancel{AB}}{ \frac{4}{ \cancel7} \: \: \cancel{AB}} }}$

$\dashrightarrow { \rm{\frac{AP}{PB} = \frac{3}{4}}}$

• Hence, the point P divides AB in the ratio of 3:4.

• Finding coordinate of p :-

• Let, ${ \rm{m_{1} = 3, \: m_{2} = 4}}$

${ \rm{x_{1} = - 2, \: x_{2} = 2}}$

${ \rm{y_{1} = - 2, \: y_{2} = - 4}}$

Using Section Formula : i.e,

${ \boxed{ \pink{ \rm{x = \frac{m_{1}x_{2}+ m_{2}x_{1}}{m_{1} + m_{2} } , {y = \frac{m_{1}y_{2}+ m_{2}y_{1}}{m_{1} + m_{2} }}}}}}$

${\dashrightarrow{ \rm{ x = \frac{3(2) + 4( - 2)}{3 + 4} , \: y = \frac{3( - 4) + 4( - 2)}{3 + 4}}}}$

${\dashrightarrow{ \rm{ x = \frac{6 - 8}{7} , \: y = \frac{ - 12 - 8}{7}}}}$

${\dashrightarrow{ \rm{ x = \frac{ - 2}{7} , \: y = \frac{ - 20}{7}}}}$

Hence, the co-ordinates of P are P(x,y) = ${ \rm{P = \frac{ - 2}{7} , \: \frac{ - 20}{7}} }$