Step-by-step explanation:

♦ Given ♦ :-

• Points P (6, 1) and Q (-2, 0)
• The ratio is 3 : 1

♦ To find ♦ :-

• The coordinates of the point A(x,y) which divides the line segment joining the points P and Q.

♦ Solution ♦ :-

Given points are : P (6, 1) and Q (-2, 0)

Let (x1, y1) = P(6,1) => x1 = 6 , y1 = 1

Let (x2, y2) = Q(-2,0) => x2 = -2 , y2 = 0

The points which divides the line segment joining the points A(x,y).

The ratio = 3 : 1

Let m1 : m2 = 3 : 1 => m1 = 3 , m2 = 1

We know that

The Section Formula : The coordinates of the point P(x,y) which divides the line segment joining the points (x1, y1) and (x2, y2) in the ratio m1:m2 internally is ( {m1x2+ m2x1}/(m1+m2) , {m1y2+m2y1}/(m1+m2) )

The coordinates of A(x,y)

= ( {(3)(-2)+(1)(6)}/(3+1) , {(3)(0)+(1)(1)}/(3+1) )

= ( {-6+6}/4 , {0+1}/4 )

= ( 0/4 , 1/4 )

= (0,1/4)

♦ Answer ♦ :-

The coordinates of the point A (x,y) which divides the given line segment joining the points P and Q is ( 0, 1/4 ).

Given :-

• The coordinates of a point A (x, y) which divides the join P (6, 1) and Q(-2, 0)

To Find :-

• In the ratio 3:1 is ?

Formula Used :-

♣️ Section formula :

[tex] \displaystyle \sf\: : \implies \: \bigg \{ \: \frac{ m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m_{2}} ,\frac{ m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m_{2}} \: \bigg \}[/tex]

Explanation :-

Given that,

[tex] \bf⇝[/tex] A = (6, 1)

[tex] \bf⇝[/tex] B = (-2, 0)

We know that,

[tex] \bf ⇢ x_{1} [/tex] = 6

[tex] \bf ⇢ x_{2} [/tex] = -2

[tex] \bf ⇢ y_{1} [/tex] = 1

[tex] \bf ⇢ y_{2} [/tex] = 0

[tex] \bf ⇢ m_{1} [/tex] = 3

[tex] \bf ⇢ m_{2} [/tex] = 1

According to the question by using section formula we get,

[tex] \displaystyle \sf\: : \implies \: \bigg \{ \: \frac{ m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m_{2}} ,\frac{ m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m_{2}} \: \bigg \}[/tex]

[tex] \displaystyle \sf\: : \implies \: \bigg \{ \: \frac{ 3 \times ( - 2) + 1 \times 6}{3 + 1} ,\frac{ 3 \times 0 + 1 \times 1 }{3 + 1} \: \bigg \}[/tex]

[tex] \displaystyle \sf\: : \implies \: \bigg \{ \: \frac{ - 6 + 6}{4} ,\frac{ 0 + 1 }{4} \: \bigg \}[/tex]

[tex] \displaystyle \sf\: : \implies \: \bigg \{ \: \frac{ 0}{4} , \: \frac{ 1 }{4} \: \bigg \}[/tex]

[tex] \displaystyle \sf\: : \implies \: \bigg \{ \: 0 , \: \frac{ 1 }{4} \: \bigg \}[/tex]

Hence,

The coordinates of a point A (x, y) which divides the line segment joining the points P and Q is [tex] \displaystyle \sf( 0, \: \frac{1}{4})[/tex]

[tex] \: [/tex]