Question

Are the lines \(l\) and \(m\) parallel? Statements: (I) \(l\) and \(m\) make equal angles with \(y=0\). (II) \(l\) and \(m\) intersect at \(x=0\).

• A. Statement (I) alone is sufficient to answer the question, but (II) alone is not sufficient.
• B. Statement (II) alone is sufficient to answer the question, but (I) alone is not sufficient.
• C. Both the statements (I) and (II) are sufficient to answer the question, but neither statement alone is sufficient.
• D. Both the statements (I) and (II) together are not sufficient to answer the question and additional data are required.

Hint:

If two lines make equal angles with the same axis, their slopes are equal. Equal slopes imply parallel lines.

Answer 1

The Correct Option is A

Concept: Two lines are parallel if and only if their slopes are equal. Also, the angle a line makes with the \(x\)-axis determines its slope: \[ m=\tan \theta \] where \(\theta\) is the angle made with the positive \(x\)-axis.

Step 1: Checking Statement (I).
Given: \[ l \text{ and } m \text{ make equal angles with } y=0 \] The equation \(y=0\) represents the \(x\)-axis. If both lines make equal angles with the \(x\)-axis, then: \[ \theta_l=\theta_m \] Thus, \[ m_l=\tan \theta_l \] and \[ m_m=\tan \theta_m \] Since: \[ \theta_l=\theta_m \] we get: \[ m_l=m_m \] Equal slopes imply the lines are parallel. So, Statement (I) alone is sufficient.

Step 2: Checking Statement (II).
Given: \[ l \text{ and } m \text{ intersect at } x=0 \] This only tells us that both lines meet somewhere on the \(y\)-axis. They may intersect and hence not be parallel. Or they could coincide. This does not determine parallelism. So, Statement (II) alone is not sufficient. Hence, Statement (I) alone is sufficient.