Question

Points P(x, 2), A(−2, 3) and B(−5, 4) are collinear.

Statement (1): Slope of PA = slope of PB = slope of AB.

Statement (2): x = 1.

Options

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Both the statements are true.

Explanation:

Given, P(x, 2), A(−2, 3) and B(−5, 4).

If three points are collinear, it means they lie on same line.

∴ Slope will be equal.

∴ Slope of PA = Slope of PB = Slope of AB

So, statement 1 is true.

Using formula,

Slope = `(y_2 − y_1)/(x_2 − x_1)`

Slope of AB = Slope of PA

⇒ `((−5) − (−2))/(4 − 3)​= (3 − 2)/((−2) − x)`

⇒ `(4 − 3)/(−5 + 2​) = (3 − 2​)/(−2 − x)`

⇒ `1/(−3) ​= 1/(−2 − x)​`

⇒ −2 − x = −3

⇒ x = 3 − 2

⇒ x = 1.

So, statement 2 is true.