Question

The point (k, 3) divides the join of (4, -3) and (8, 5) in a certain ratio. What is the value of k?
1 3
2 5
3 7
4 (We cannot find k without knowing the ratio of division.)

Answer 1

Answer:

Question :-

• The point (k, 3) divides the join of (4, -3) and (8, 5) in a certain ratio. What is the value of k?

Answer :-

Given :-

• The point (k, 3) divides the join of (4, -3) and (8, 5) in a certain ratio.

To Find :-

• The value of 'k'.

Formula Used :-

Section formula is used to determine the coordinate of a point that divides a line segment joining two points into two parts such that the ratio of their length is m:n.

Let P and Q be the given two points (x1,y1) and (x2,y2) respectively, and M be the point dividing the line-segment PQ internally in the ratio m:n, then form the sectional formula for determining the coordinate of a point M is given by:

$\large M (x,y) = \left ( \frac{mx_{2}+nx_{1}}{m+n} , \frac{my_{2}+ny_{1}}{m+n} \right )$

Solution:-

Let the point M(k, 3) divides the join of P(4, -3) and Q(8, 5) in the ration p : 1.

So, using section Formula, the coordinates of M divides the line segment joining P and Q is given by

$\large M (x,y) = \left ( \frac{mx_{2}+nx_{1}}{m+n} , \frac{my_{2}+ny_{1}}{m+n} \right )$

On substituting the values, m = p, n = 1, x = k, y = 3,

$\bf \:x_1 = 4, \: y_1 = -3, \: x_2=8, \: y_2=5$

we get as

$\bf\implies \:(k ,3 ) \: = ( \dfrac{4 \times 1 + 8 \times p}{p + 1} , \dfrac{ - 3 \times 1 + 5 \times p}{p + 1} )$

$\bf\implies \:(k ,3 ) \: = ( \dfrac{8 p + 4}{p + 1} , \dfrac{ - 3 + 5 p}{p + 1} )$

On comparing, we get

$\bf\implies \:3 = \dfrac{ - 3 + 5p}{p + 1}$

$\bf\implies \:3p + 3 = 5p - 3$

$\bf\implies \:3p - 5p = - 3 - 3$

$\bf\implies \: - 2p = - 6$

$\bf\implies3 \:{\cancel\dfrac{ - 6}{ - 2}=p} \:$

$\bf\implies \:p = 3$

$\bf\implies \:k = \dfrac{8p + 4}{p + 1}$

$\bf\implies \:k = \dfrac{8 \times 3 + 4}{3 + 1}$

$\bf\implies \:k = \dfrac{28}{4} = 7$

$\bf\implies \:The \: value \: of \: k = 7$

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