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If $P − 2, 4 and Q − 2, − 5$ then find the coordinates of points I and J such that $P I = I J = J Q$ .

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I − 2, 1 and J − 2, − 2

I − 2, 4 and J − 2, − 5

I − 1, 4 and J − 2, − 5

I − 2, 4 and J 2, − 5

answer is A.

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Detailed Solution

Given points,

P − 2, 4 and Q − 2, − 5

.
We have to find the coordinates of points I and J such that

P I = I J = J Q

.
Since,

P I = I J = J Q

\Rightarrow

the points I and J are on the line of PQ.
Now,
Point I divide PQ in a ratio of 1:2.
We know that,
Section Formula for the coordinates

(x, y)

of a point O that break the line segment joining the points

A (x_{1}, y_{1}) and B (x_{2}, y_{2})

in the ratio

m_{1} : m_{2}

is given by,

x = \frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}}

and

y = \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}}

.
Then, the coordinates of the point I is,

− 2 − 4 3, − 5 + 8 3 = (− 2, 1)

.
And
The point J divides PQ in a ratio of 1:2.
Then, the coordinates of point J are,

− 4 − 2 3, − 10 + 4 3 = (− 2, − 2)

.
Therefore, the coordinates of the point I is (-2,1), and the coordinates of point J is (-2, -2).
Hence, option (1) is correct.

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