IPMAT Indore 2025 (MCQ) - Let A(1,3) and B(5,1) be two points. If a line with slope m intersects AB at an angle of 45°, then the possible values of m are | PYQs + Solutions

IPMAT Indore 2025

Geometry

Straight Lines

Medium

Let $A(1,3)$ and $B(5,1)$ be two points. If a line with slope $m$ intersects $AB$ at an angle of $45°$ , then the possible values of $m$ are

7, \frac{1}{7}

3, \frac{1}{3}

-3, \frac{1}{3}

5, -\frac{1}{5}

✅ Correct Option: 3

Calculate the slope of line

AB

m_{AB} = \dfrac{y_B - y_A}{x_B - x_A} = \dfrac{1 - 3}{5 - 1} = \dfrac{-2}{4} = -\dfrac{1}{2}

When two lines with slopes

m_1

and

m_2

intersect at angle

θ

, we use the formula:

\tan θ = \left|\dfrac{m_2 - m_1}{1 + m_1m_2}\right|

In our case,

m_1 = -\dfrac{1}{2}

and

m_2 = m

. Since the angle is

45°

\tan(45°) = 1

So we have:

1 = \left|\dfrac{m - (-\frac{1}{2})}{1 + (-\frac{1}{2})m}\right| = \left|\dfrac{m + \frac{1}{2}}{1 - \frac{m}{2}}\right|

Removing the absolute value gives us two cases:

Case 1:

\dfrac{m + \frac{1}{2}}{1 - \frac{m}{2}} = 1

m + \dfrac{1}{2} = 1 - \dfrac{m}{2}

m + \dfrac{m}{2} = 1 - \dfrac{1}{2}

\dfrac{3m}{2} = \dfrac{1}{2}

m = \dfrac{1}{3}

Case 2:

\dfrac{m + \frac{1}{2}}{1 - \frac{m}{2}} = -1

m + \frac{1}{2} = -1 + \frac{m}{2}

m - \frac{m}{2} = -1 - \frac{1}{2}

\frac{m}{2} = -\frac{3}{2}

m = -3

Therefore, the possible values of

m

are

\dfrac{1}{3}

and

-3