Assertion (A) : y = x + 4 and y = $\sqrt{3}x + 5$ are two intersecting lines.

Reason (R) : The inclinations of both the given lines are not equal.

1. A is true, R is false

2. A is true, R is true

3. A is false, R is true

4. A is false, R is false

Answer

Given,

1st equation :

y = x + 4

Comparing above equation with y = mx + c, we get :

m = 1

Slope of 1st line = 1.

2nd equation :

y = $\sqrt{3}x + 5$

Comparing above equation with y = mx + c, we get :

m = $\sqrt{3}$

Slope of 2nd line = $\sqrt{3}$.

Equating equations (1) and (2), we get :

$\Rightarrow x + 4 = \sqrt{3}x + 5 \\[1em] \Rightarrow x - \sqrt{3}x = 5 - 4 \\[1em] \Rightarrow x(1 - \sqrt{3}) = 1 \\[1em] \Rightarrow x = \dfrac{1}{1 - \sqrt{3}}$

Rationalising,

$\Rightarrow x = \dfrac{1}{1 - \sqrt{3}} \times \dfrac{1 + \sqrt{3}}{1 + \sqrt{3}} \\[1em] \Rightarrow x = \dfrac{1 + \sqrt{3}}{1^2 - (\sqrt{3})^2} \\[1em] \Rightarrow x = \dfrac{1 + \sqrt{3}}{1 - 3} \\[1em] \Rightarrow x = \dfrac{1 + \sqrt{3}}{-2} \\[1em] \Rightarrow x = \dfrac{-(1 + \sqrt{3})}{2}$

Substituting value of x in equation (1), we get :

$\Rightarrow y = \dfrac{-(1 + \sqrt{3})}{2} + 4 \\[1em] \Rightarrow y = \dfrac{-1 - \sqrt{3} + 8}{2} \\[1em] \Rightarrow y = \dfrac{7 - \sqrt{3}}{2}.$

∴ y = x + 4 and y = $\sqrt{3}x + 5$ are two intersecting lines and also the inclinations of both the given lines are not equal.

Hence, Option 2 is the correct option.