###### Advertisements

###### Advertisements

Find the ratio in which the point P(*x*, 2) divides the line segment joining the points A(12, 5) and B(4, −3). Also, find the value of *x*.

###### Advertisements

#### Solution 1

Let the point P (*x*, 2) divide the line segment joining the points A (12, 5) and B (4, −3) in the ratio *k:*1.

Then, the coordinates of P are `((4k+12)/(k+1),(-3k+5)/(k+1))`

Now, the coordinates of P are (*x*, 2).

`therefore (4k+12)/(k+1)=x and (-3k+5)/(k+1)=2`

`(-3k+5)/(k+1)=2`

`-3k+5=2k+2`

`5k=3`

`k=3/5`

Substituting `k=3/5 " in" (4k+12)/(k+1)=x`

we get

`x=(4xx3/5+12)/(3/5+1)`

`x=(12+60)/(3+5)`

`x=72/8`

x=9

Thus, the value of *x *is 9.

Also, the point P divides the line segment joining the points A(12, 5) and (4, −3) in the ratio 3/5:1 i.e. 3:5.

#### Solution 2

Let k be the ratio in which the point P(x,2) divides the line joining the points

`A(x_1 =12, y_1=5) and B(x_2 = 4, y_2 = -3 ) .` Then

`x= (kxx4+12)/(k+1) and 2 = (kxx (-3)+5) /(k+1)`

Now,

` 2 = (kxx (-3)+5)/(k+1) ⇒ 2k+2 = -3k +5 ⇒ k=3/5`

Hence, the required ratio is3:5 .

#### APPEARS IN

#### RELATED QUESTIONS

If the coordinates of the mid points of the sides of a triangle are (1, 1), (2, – 3) and (3, 4) Find its centroid

Find the ratio in which the line segment joining the points (-3, 10) and (6, - 8) is divided by (-1, 6).

Find the ratio in which the line segment joining A (1, − 5) and B (− 4, 5) is divided by the x-axis. Also, find the coordinates of the point of division.

Find the ratio in which P(4, *m*) divides the line segment joining the points A(2, 3) and B(6, –3). Hence find *m*.

Three vertices of a parallelogram are (a+b, a-b), (2a+b, 2a-b), (a-b, a+b). Find the fourth vertex.

Calculate the ratio in which the line joining the points (-3, -1) and (5, 7) is divided by the line x = 2. Also, find the co-ordinates of the point of intersection.

Calculate the ratio in which the line joining A(6, 5) and B(4, -3) is divided by the line y = 2.

The line joining P(-4, 5) and Q(3, 2) intersects the y-axis at point R. PM and QN are perpendicular from P and Q on the x-axis Find:

(i) the ratio PR : RQ

(ii) the coordinates of R.

(iii) the area of the quadrilateral PMNQ.

In the given figure line APB meets the x-axis at point A and y-axis at point B. P is the point (-4,2) and AP : PB = 1 : 2. Find the co-ordinates of A and B.

Find the length of the hypotenuse of a square whose side is 16 cm.

Find the ratio in which the line y = -1 divides the line segment joining (6, 5) and (-2, -11). Find the coordinates of the point of intersection.

The origin o (0, O), P (-6, 9) and Q (12, -3) are vertices of triangle OPQ. Point M divides OP in the ratio 1: 2 and point N divides OQ in the ratio 1: 2. Find the coordinates of points M and N. Also, show that 3MN = PQ.

Find the ratio in which the point R ( 1, 5) divides the line segment joining the points S (-2, -1) and T (5, 13).

The points A, B and C divides the line segment MN in four equal parts. The coordinates of Mand N are (-1, 10) and (7, -2) respectively. Find the coordinates of A, B and C.

Find the ratio in which the line segment joining P ( 4, -6) and Q ( -3, 8) is divided by the line y = 0.

In the figure given below, the line segment AB meets X-axis at A and Y-axis at B. The point P (- 3, 4) on AB divides it in the ratio 2 : 3. Find the coordinates of A and B.

Using section formula, show that the points A(7, −5), B(9, −3) and C(13, 1) are collinear

If point P(1, 1) divide segment joining point A and point B(–1, –1) in the ratio 5 : 2, then the coordinates of A are ______

If point P divides segment AB in the ratio 1 : 3 where A(– 5, 3) and B(3, – 5), then the coordinates of P are ______

The point Q divides segment joining A(3, 5) and B(7, 9) in the ratio 2 : 3. Find the X-coordinate of Q

Point P(– 4, 6) divides point A(– 6, 10) and B(m, n) in the ratio 2:1, then find the coordinates of point B

Find the ratio in which Y-axis divides the point A(3, 5) and point B(– 6, 7). Find the coordinates of the point

The perpendicular bisector of the line segment joining the points A(1, 5) and B(4, 6) cuts the y-axis at ______.

If the points A(1, 2), O(0, 0), C(a, b) are collinear, then ______.

The vertices of a parallelogram in order are A(1, 2), B(4, y), C(x, 6) and D(3, 5). Then (x, y) is ______.

The points A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) are the vertices of ∆ABC. What are the coordinates of the centroid of the triangle ABC?

Complete the following activity to find the coordinates of point P which divides seg AB in the ratio 3:1 where A(4, – 3) and B(8, 5).

Activity:

∴ By section formula,

∴ x = `("m"x_2 + "n"x_1)/square`,

∴ x = `(3 xx 8 + 1 xx 4)/(3 + 1)`,

= `(square + 4)/4`,

∴ x = `square`,

∴ y = `square/("m" + "n")`

∴ y = `(3 xx 5 + 1 xx (-3))/(3 + 1)`

= `(square - 3)/4`

∴ y = `square`

Point C divides the line segment whose points are A(4, –6) and B(5, 9) in the ratio 2:1. Find the coordinates of C.

In what ratio does the Y-axis divide the line segment P(– 3, 1) and Q(6, 2)?

A line intersects y-axis and x-axis at point P and Q, respectively. If R(2, 5) is the mid-point of line segment PQ, them find the coordinates of P and Q.