JEE Main Mathematics: Lines through the point of intersection of two given lines Practice Questions
8 free multiple-choice questions on Lines through the point of intersection of two given lines (Analytical Geometry) for JEE Main Mathematics, each with the correct answer and a step-by-step explanation. Attempt a question, then reveal the solution to check your reasoning.
Practice thousands more JEE Main questions in the app →Q1. Two lines L₁ and L₂ intersect at point P. If a third line L₃ also passes through P, what property does L₃ have with respect to L₁ and L₂ at point P?
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Correct answer: C — It forms an angle θ with both L₁ and L₂.
If three distinct lines (L₁, L₂, L₃) intersect at a single point P, the lines form various angles. While L₃ is not necessarily the angle bisector, it simply forms angles with L₁ and L₂ at P. The most general statement is that L₃ passes through the intersection point P.
Q2. Two lines intersect at point P. If the measure of one angle formed by these lines is 60°, what is the measure of the adjacent angle?
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Correct answer: B — 120°
When two lines intersect, the angles formed on a straight line are supplementary (add up to 180°). Adjacent angles formed by intersecting lines on a straight line sum up to 180°. Therefore, 180° - 60° = 120°.
Q3. Two lines intersect at a point P. If the angles formed by the intersection are θ and 180° - θ, what is the relationship between any line passing through P and the lines themselves?
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Correct answer: B — It forms an angle θ with both lines.
Any line passing through the point of intersection P forms an angle with the two intersecting lines. If one angle is θ, the adjacent angle formed by this new line and one of the original lines will be θ (if the line lies within the angle), or 180° - θ (if the line lies outside the angle). In general, a line passing through P forms angles with the original lines.
Q4. If two lines intersect at point P, and the angles formed are θ and 180° - θ, what is the relationship between θ and its supplement?
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Correct answer: A — They are supplementary
Angles that form a linear pair (angles on a straight line) are supplementary, meaning their sum is 180°. If θ is one angle, its supplement is 180° - θ.
Q5. If two lines L₁ and L₂ intersect at point A, how many distinct lines can pass through the point A?
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Correct answer: A — Infinitely many
Through any single point in a plane, there are infinitely many lines that can be drawn. Since the two lines L₁ and L₂ intersect at A, any line passing through A is a valid line segment, and there are infinitely many such lines.
Q6. Consider two intersecting lines. If a line is drawn through their point of intersection, this line defines the relationship between the lines. If the lines are not parallel, what is the consequence of drawing a line through their intersection point?
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Correct answer: B — The line passes through the point of intersection, establishing a relationship based on the angles formed.
The fundamental relationship established when drawing a line through the intersection point of two lines is that this new line forms angles with the original lines, directly relating the geometric configuration of the intersecting lines at that specific point.
Q7. Two lines l₁ and l₂ intersect at P. If a line l₃ passes through P, the angle bisector of the angle formed by l₁ and l₂ divides the angle into which ratio?
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Correct answer: A — 1:1
The angle bisector divides the angle into two equal halves. Thus, it divides the angle into the ratio 1:1.
Q8. Three lines L₁, L₂, L₃ intersect pairwise at points A, B, and C, respectively. If the angle formed at the intersection point P (where L₁ and L₂ meet) is 60°, and L₃ passes through P, what is the sum of the angles formed by L₃ with L₁ and L₂?
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Correct answer: A — 180^°
When three lines intersect at a single point P, the angles formed around that point sum up to 360°. If L₁ and L₂ form an angle of 60°, the remaining angle formed by L₃ is 360° - 60° = 300°. The question asks for the sum of the angles formed by L₃ with L₁ and L₂ (meaning the angles formed by the rays emanating from P), which defines the total plane around the intersection. The angles formed are 60°, 300°, and potentially their supplements, but the total angle around the point is 360°. The required sum of the adjacent angles formed by the three lines is 360°. If the question refers specifically to the angle between L₁ and L₂ (60 degrees), the angle between L₁ and L₃ and L₂ and L₃ must sum up to the total angle around P, which is 360°.
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