In what ratio is the line segment joining the points (3,

Q) In what ratio is the line segment joining the points (3, – 5) and (- 1, 6) divided by the line y = x ?

Ans:

Step 1: Now, by section formula, coordinates of point P (X, Y) which lies between two points (x₁, y₁), (x₂, y₂) will be given by:

P (X,Y) = $(\frac{m_1 \times_2 + m_2 \times_1}{m_1 + m_2}, \frac{m_1 Y_2 + m_2 Y_1}{m_1 + m_2})$

here, point divides the line in ratio of m_{1 :} m₂

Step 2: Now if the given points A (3, -5) and B(- 1, 6) are divided in the ratio of m : n, then:

coordinates of dividing point (x, y) = $(\frac{m (- 1) + n (3)}{m + n}, \frac{m (6) + n(- 5)}{m + n})$

= $(\frac{(- m + 3 n)}{m + n}, \frac{(6 m - 5 n)}{m + n})$

Step 3: Next, since this point lies on line y = x, this point will satisfy the equation

∴ $\frac{(6 m - 5 n)}{m + n} = \frac{(- m + 3 n)}{m + n}$

∴ 6 m – 5 n = 3 n – m

∴ 7 m = 8 n

∴ m : n = 8 : 7

Therefore, the line segment divides the line in ratio of 8:7.

Check: Let’s plot the points on the graph:

Here, we can see that line x = y is cutting line AB in a ratio where m is almost equal to n. hence our answer is correct.

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