Question

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.

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

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:

⇒ \[x + 4 = \sqrt{3}x + 5\]

⇒ \[x - \sqrt{3}x = 5 - 4\]

⇒ \[x(1 - \sqrt{3}) = 1\]

⇒ \[x = \frac{1}{1 - \sqrt{3}}\]

Rationalising,

⇒ \[x = \frac{1}{1 - \sqrt{3}} \times \frac{1 + \sqrt{3}}{1 + \sqrt{3}}\]

⇒ \[x = \frac{1 + \sqrt{3}}{1^2 - (\sqrt{3})^2}\]

⇒ \[x = \frac{1 + \sqrt{3}}{1 - 3}\]

⇒ \[x = \frac{1 + \sqrt{3}}{-2}\]

⇒ \[x = \frac{-(1 + \sqrt{3})}{2}\]

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

⇒ \[y = \frac{-(1 + \sqrt{3})}{2} + 4\]

⇒ \[y = \frac{-1 - \sqrt{3} + 8}{2}\]

⇒ \[y = \frac{7 - \sqrt{3}}{2}.\]

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