Math, asked by saipushkarini, 10 months ago

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​

Answers

Answered by BrainlyConqueror0901
10

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 }

Answered by Anonymous
5

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 ☺️

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