In what ratio does the point P 3 and 4 divide the line segment joining the point a 1 and 2 and B 6 and 7?

Question 52 Section Formula Exercise 11

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Answer:

Solution:

In coordinate geometry, the Section formula is used to determine the internal or external ratio at which a line segment is divided by a point.

Let the ratio that the point (-4,b) divide the line segment joining the points (2,-2) and (-14,6) be m:n,

Here x1 = 2 , y1 = -2 , x2 = -14, y2 = 6, x = -4, y = b

By section formula,

x=\frac{\left(mx_2+nx_1\right)}{(m+n)}\\-4=\frac{\left(m\times-14+n\times2\right)}{(m+n)}\\-4=\frac{\left(-14m+2n\right)}{(m+n)}\\-4(m+n)=-14m+2n\\-4m-4n=-14m+2n\\-4m+14m=2n+4n\\ 10m=6n\\ \frac{m}{n}=\frac{6}{10}=\frac{3}{5}

Hence the ratio m:n is 3:5.

By Section formula,

y=\frac{\left(my_2+ny_1\right)}{\left(m+n\right)}\\b=\frac{\left(3\times6+5\times-2\right)}{(3+5)}\\b=\frac{\left(18-10\right)}{8}\\b=\frac{8}{8}\\b=1

Hence the value of b is 1 and the ratio m:n is 3:5.

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Solution:

The coordinates of the point P(x, y) which divides the line segment joining the points A(x₁, y₁) and B(x₂, y₂), internally, in the ratio m₁: m₂ is given by the Section Formula: P(x, y) = [(mx₂ + nx₁) / m + n, (my₂ + ny₁) / m + n]

Let the ratio in which the line segment joining A(- 3, 10) and B(6, - 8) be divided by point C(- 1, 6) be k : 1.

By Section formula, C(x, y) = [(mx₂ + nx₁) / m + n, (my₂ + ny₁) / m + n]

m = k, n = 1

Therefore,

- 1 = (6k - 3) / (k + 1)

- k - 1 = 6k - 3

7k = 2

k = 2 / 7

Hence, the point C divides line segment AB in the ratio 2 : 7.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 7

Video Solution:

NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.2 Question 4

Summary:

The ratio in which the line segment joining the points (- 3, 10) and (6, - 8) is divided by (- 1, 6) is 2 : 7.

☛ Related Questions:

• 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.
• If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
• Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, - 3) and B is (1, 4).
• If A and B are (- 2, - 2) and (2, - 4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.

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