Question

find K so that the point P (- 4,6) lines on line segment joining A (k,10)B(3, - 8) also find ratio in which P divides the line segmentAB​

Answer 1

Answer:

${\bold{\therefore Ratio=2:7}}$

${\bold{\therefore k=-6}}$

Step-by-step explanation:

${\bold{\huge{\underline{SOLUTION-}}}}$

• In the given question information given about a line segment AB whose Coordinate of A and B is given. P is the point on the line segment whose coordinates are given.

• We have to find the ratio in which point P divides the line segment and value of k.

$\underline \bold{Given : } \\ \implies Coordinate \: of \: P = ( - 4,6) \\ \implies Coordinate \: of \: A = (k,10) \\ \implies Coordinate \: of \: B= (3,- 8) \\ \implies Let \: Ratio = (n : 1) \\ \\ \underline \bold{To \: Find : } \\ \implies k = ? \\ \implies Ratio = ?$

• According to given question :

$\implies y = \frac{n y_{2} + 1 y_{1} }{n + 1} \\ \implies 6 = \frac{n \times - 8 + 1 \times 10}{n + 1} \\ \implies 6n + 1 = - 8n + 10 \\ \implies 6n + 8n = 10 - 6 \\ \implies 14n = 4 \\ \implies n = \frac{4}{14} \\ \bold{\implies n = \frac{2}{7} } \\ \\ \bold{\therefore Ratio = \frac{n}{1} = \frac{2}{7} = 2 : 7} \\ \\ \bold{By \: mid \: point \: formula : } \\ \bold{For \: x \: abssisa} \\ \implies x = \frac{n x_{2} + 1 x_{1} }{n + 1} \\ \implies - 4 = \frac{2 \times 3 + 7 \times k}{2 + 7} \\ \implies - 4 \times 9 = 6+ 7k \\ \implies - 36 - 6 = 7k \\ \implies - 42 = 7k \\ {\implies k = \frac{ - 42}{7} } \\ \bold{\implies k = - 6 }$

Answer 2

ANSWER:-

Given:

The given points are:

⏺️P(-4,6)

⏺️A(k,10)

⏺️B(3,-8).

To find:

Find the value of k & find the ratio in which P divides the line segment AB.

Solution:

Since, the points P lies on the line segment joining the points A & B, so A, P & B are collinear.

We know that points, (x1,y1);(x2,y2)& (x3,y3) are collinear.

Therefore,

$\frac{1}{2} [x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)] = 0 \\ Since, \: A \: P\: and \: B \: are \: collinear \: points.\\ \\ = > [k(6 + 8) - 4( - 8 - 10) + 3(10 - 6)] = 0 \\ \\ = > [(k(14) - 4( - 18) + 3(4)]= 0 \\ \\ = > 14k + 72 + 12 = 0 \\ \\ = > 14k + 84 = 0 \\ \\ = > 14k = - 84 \\ \\ = > k = \frac{ - 84}{14} \\ \\ = > k = - 6$

Now,

Coordinates of P are: P(-4,6)

Let P divides AB in the ratio of m:1.

Therefore,

By section formula;

$= > - 4 = \frac{3m - 6}{m + 1} \\ \\ = > - 4m - 4 = 3m - 6 \\ \\ = > - 4 m - 3m = -6+4 \\ \\ = > - 7m = -2 \\ \\ = > m = \frac{2}{7}$

So,

P divides AB externally in the ratio of 2:7.

Hope it helps ☺️