Question

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

Answer 1

Answer :

The required coordinates of P is (1/2 , -11/4)

Given :

• The points are A(-2 , -2) and B(2 , -4)
• P is a point on AB such that AP = (3/7)AB

To Find :

• The coordinates of P

Formula to be used :

If a point (x , y) divides a line segment joining the points (x₁ , y₁) and (x₂ , y₂) in the ratio m:n then x and y are given by :

$\sf \star \: \: x = \dfrac{mx_{2}+nx_{1}}{m+n} \: \: , y = \dfrac{my_{1}+ny_{2}}{m+n}$

Solution :

Let the coordinates of P be (x , y)

By question we get :

$\sf \implies AP = \dfrac{3}{7}AB \\\\ \sf \implies 7AP = 3AB ....... (1)$

Since the point P lies on AB so ,

$\sf \implies AP + BP = AB \\\\ \sf Multiplying \: \: both \: \: sides \: \: by \: 3 \\\\ \sf \implies 3AP + 3BP = 3AB \\\\ \sf \implies 3AP + 3BP = 7AP \{ From \: \: (1) \} \\\\ \sf \implies 3BP = 5AP \\\\ \sf \implies \dfrac{3}{5} = \dfrac{AP}{BP} \\\\ \sf \implies AP:BP = 3:5$

Applying section formula we have :

For X-coordinate of point P

$\sf \implies x = \dfrac{3\times 2 + 5\times (-2)}{3+5} \\\\ \sf \implies x = \dfrac{6 - 10}{8} \\\\ \sf \implies x = \dfrac{4}{8} \\\\ \sf \implies x = \dfrac{1}{2}$

And for Y-coordinate of point P

$\sf \implies y = \dfrac{3\times (-4) + 5\times (-2) }{3+5} \\\\ \sf \implies y = \dfrac{-12 -10}{8} \\\\ \sf \implies y = \dfrac{-22}{8} \\\\ \sf \implies y =\dfrac{-11}{4}$

$\sf Hence \: the \: coordinates \: of \: P \: is \: (\dfrac{1}{2} , \dfrac{-11}{4} )$

Answer 2

$\rule{200}2$

$\huge\tt{QUESTION:}$

• If A and B (-2,-2) and (2,-4) respectively
• find the co ordinates of P such that AP =3/7AB and P lies on the line segment AB

$\rule{200}2$

$\huge\tt{SOLUTION:}$

Let the coordinates of p be (x,y)

↪AP = 3/7 AP

↪7AP = 3AB __(EQ.1)

SINCE, P LIES ON AB,

↪AP + BP = AB

↪3AP + 3BP = 3AB

↪3AP + 3BP = 7AP

↪3BP = 5AP

↪AP : BP = 3:5

$\rule{200}2$

For x cordinate of point p,

↪x = 3 × 2 + 5 × (-2) / 3 + 5

↪x = 6 - 10 / 8

↪x = 4 / 8 or 1 / 2

For y cordinate of point p,

↪y = 3 × (-4) + 5 × (-2) / 3 + 5

↪y = -12 - 10

↪y = -22 / 8 or -11 / 4

$\rule{200}2$