Find the ratio in which point C(3,3) divides the line joining A(7,1) and (1,4)?

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Answer

Answer

Ratio in which point C(3,3) divides the line joining A(7,1) and (1,4)

Let's solve this problem using the section formula.

Section Formula

The section formula is used to find the coordinates of a point that divides a line segment into two parts in a given ratio. The formula is:

Let A(x1, y1) and B(x2, y2) be the endpoints of a line segment. Let P(x, y) be a point that divides AB in the ratio m:n. Then, the coordinates of P are:

x = (nx2 + mx1) / (m + n)

y = (ny2 + my1) / (m + n)

Applying Section Formula to Find Ratio

Let's apply the section formula to find the ratio in which point C divides the line joining A and B.

Given:

A(7, 1), B(1, 4), C(3, 3)

Let's assume that point C divides the line AB in the ratio m:n.

Using the section formula:

x = (nx2 + mx1) / (m + n)

y = (ny2 + my1) / (m + n)

Substituting the given values:

3 = (n1 + m7) / (m + n)

3 = (n4 + m1) / (m + n)

Multiplying the first equation by 4 and the second equation by 7:

12 = 4n + 28m / (m + n)

21 = 7n + 4m / (m + n)

Adding the two equations:

33 = 11n + 32m / (m + n)

Multiplying both sides by (m + n):

33(m + n) = 11n + 32m

Expanding:

33m + 33n = 11n + 32m

Collecting like terms:

33n - 11n = 32m - 33m

22n = -m

Since m and n are in the ratio m:n, we can write:

m = 22k, n = -k

where k is a constant.

Substituting these values in the first equation:

3 = (-k1 + 22k7) / (22k - k)

Simplifying:

3 = (153k) / 21k

3 = 7.29

This is a contradiction, so our assumption that point C divides the line AB in the ratio m:n is wrong.

Therefore, point C does not divide the line AB in a ratio of integers.

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