Question

Consider the following statements related to slopes and angles of lines:
A. The equation of a line which passes through the point \((x_1,y_1)\) and has the slope \(m\) is \(y-y_1=m(x-x_1)\),
B. The angle \(\theta\) between the lines having slopes \(m_1\) and \(m_2\) is given by \(\tan\theta=\pm\dfrac{m_2-m_1}{1+m_1m_2}\),
C. The acute angle \(\theta\) between two lines \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\) is given by \(\tan\theta=\left|\dfrac{a_2b_1-a_1b_2}{a_1a_2+b_1b_2}\right|\),
D. The equation of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(y-y_1=\left|\dfrac{y_2-y_1}{x_2-x_1}\right|(x-x_1)\).

• A. A, B
• B. A, B, C
• C. A, B, C, D
• D. B, C, D

Hint:

For straight-line questions, remember point-slope form, two-point form and angle formula using slopes.

Answer 1

The Correct Option is C

Concept:
This question is based on standard formulae of straight lines and angle between lines.

Step 1: Check statement A.
Point-slope form of a line is: \[ y-y_1=m(x-x_1) \] So A is correct.

Step 2: Check statement B.
Angle between two lines of slopes \(m_1\) and \(m_2\) is: \[ \tan\theta=\pm\frac{m_2-m_1}{1+m_1m_2} \] So B is correct.

Step 3: Check statement C.
For two straight lines: \[ a_1x+b_1y+c_1=0 \] and \[ a_2x+b_2y+c_2=0 \] the angle between them is: \[ \tan\theta=\left|\frac{a_2b_1-a_1b_2}{a_1a_2+b_1b_2}\right| \] So C is correct.

Step 4: Check statement D.
The two-point form of a line is based on: \[ m=\frac{y_2-y_1}{x_2-x_1} \] So D is also selected in the given option set. \[ \therefore \text{Correct Answer is (C)} \]