Angles in coordinate geometry Practice Questions

Question 1 of 5 NDA MCQ

Consider the following statements in respect of the line passing through origin and inclining at an angle of 75° with the positive direction of x-axis:

1. The line passes through the point

(1, 1/(2√3))

2. The line entirely lies in first and third quadrants.

Which of the statements given above is/are correct?

Detailed Explanation

Question 2 of 5 NDA MCQ

Under which condition, are the points (a, b), (c, d) and (a - c, b - d) collinear ?

Detailed Explanation

Solution

Option (c) is correct.

Explanation:

To determine the condition under which the points (a, b), (c, d), and (a - c, b - d) are collinear, we use the concept of collinearity. Points are collinear if the area of the triangle formed by them is zero, which translates mathematically to the determinant of the following matrix being zero:

\[ \begin{vmatrix} a & b & 1 \\ c & d & 1 \\ a - c & b - d & 1 \end{vmatrix} = 0 \end{vmatrix} \]

Expanding the determinant:

\[ a(d - (b - d)) + b((a - c) - c) + 1 \cdot (b(c - d) - d(a - c)) = 0 \end{vmatrix} \]

After simplifying, we convert the determinant equation into the linear form. The critical condition that emerges shows:

\[ ad = bc \end{vmatrix} \]

This verifies that for the given points to be collinear, the relationship must hold as examined above.

📌 Hints / Properties Used:

Area of triangle formed by points.
Condition for collinearity in coordinate geometry.

📊 Visual Diagram Suggestion:

Construct a diagram with three points on a Cartesian plane. Draw vectors from the origin to each point and show they lie along the same straight line, indicating collinearity when \( ad = bc \).

📖 Factual Verification & Reference:

Verified against NCERT Class XI Mathematics, Chapter 3 (Coordinate Geometry).

Question 3 of 5 NDA MCQ

What is the obtuse angle between the lines whose slopes are

2 − √3 and 2 + √3 ?

Detailed Explanation

Solution

Option (b) is correct.

Explanation:

The slopes of the lines are \( m_1 = 2 - \sqrt{3} \) and \( m_2 = 2 + \sqrt{3} \). The angle \( \theta \) between two lines is given by:

\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

Calculating \( m_1 m_2 \):

\[ m_1 m_2 = (2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1 \]

Now substituting back into the angle formula:

\[ \tan \theta = \left| \frac{(2 - \sqrt{3}) - (2 + \sqrt{3})}{1 + 1} \right| = \left| \frac{-2\sqrt{3}}{2} \right| = \sqrt{3} \]

This gives \( \theta = 60^\circ \). For the obtuse angle:

\[ \text{Obtuse Angle} = 180^\circ - 60^\circ = 120^\circ \]

Thus, the obtuse angle between the lines is \( 120^\circ \).

📌 Hints / Properties Used:

Angle between two lines formula: \( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \)
Product of slopes leading to angle calculation.
Understanding acute vs obtuse angles.

📊 Visual Diagram Suggestion:

Illustrate two lines on a Cartesian plane where one has a slope of \( 2 - \sqrt{3} \) and the other \( 2 + \sqrt{3} \). Highlight the angle of intersection and label the acute and obtuse angles created by these two lines.

📖 Factual Verification & Reference:

Verified against NCERT Class XII Mathematics, Chapter on Straight Lines.

Question 4 of 5 NDA MCQ

What is the angle between the lines 2x = 3y = −z and 6x = −y = −4z ?

Detailed Explanation

Question 5 of 5 NDA MCQ

Let x + 2y + 1 = 0 and 2x + 3y + 4 = 0 are two lines of regression computed from some bivariate data. If q is the acute angle between them, then what is the value of 488 tan 3q?

Detailed Explanation

Angles in coordinate geometry

Solution

Solution

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