Find the ratio in which the point (- 1, k) divides the line segment

Q) Find the ratio in which the point (- 1, k) divides the line segment joining the points (- 3, 10) and (6, – 8). Also, find the value of k.

Ans:

Step 1: Finding the division ratio:

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

Let’s consider the point C(-1, k) divides the line AB in ratio of P : 1 and points given are (- 3, 10) and (6, – 8).

By transferring values in the above section formula, we get:

For x coordinate:

∴ – 1 = $\frac{P \times 6 + 1 \times (- 3)}{P + 1}$

∴ (- 1) (P + 1) = 6 P – 3

∴ – P – 1 = 6 P – 3

∴ 7 P = 3 – 1 = 2

∴ P = $\frac{7}{2}$

∴ P : 1 = 7 : 2

Therefore, the point C divides the line AB in ratio of 7:2.

Step 2: Finding value of k:

Let’s find value of y coordinate from section formula:

y = $\frac{P \times (- 8) + 1 \times 10}{P + 1}$

∴ k = $\frac{10 - 8 P}{P + 1}$

∴ k (P + 1) = 10 – 8 P

By substituting value of P = $\frac{7}{2}$ , we get:

k ( $\frac{7}{2}$ + 1) = 10 – 8 x $\frac{7}{2}$

∴ k ( $\frac{9}{2}$ ) = 10 – 28 = – 18

∴ k = – 18 x $\frac{2}{9}$ = – 4

Therefore, value of y is – 4.

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