IPMAT Indore 2025 · Quantitative Ability · Question 24

IPMAT Indore 2025 previous year question on Coordinate Geometry (Basics), with the answer and a step-by-step solution.

Coordinate Geometry (Basics)Geometry and MensurationMultiple choiceMedium
Let A\left( 1,3 \right) and B\left( 5,1 \right) be two points. If a line with slope m intersects AB at an angle of {45}^{∘}, then the possible values of m are
  1. A
    7,\frac{1}{7}
  2. B
    3,\frac{1}{3}
  3. C
    -3,\frac{1}{3}
  4. D
    5,-\frac{1}{5}
Pick an option, then check.
Answer & solution

Answer: Option C

Step 1 — slope of AB.

{m}_{1}=\frac{1-3}{5-1}=\frac{-2}{4}=-\frac{1}{2}

Step 2 — the angle-between-two-lines formula.

\tan \theta =\left| \frac{m-{m}_{1}}{1+m\,{m}_{1}}\right|

With \theta ={45}^{∘}, \tan \theta =1, so

\left| \frac{m+1/2}{1-m/2}\right|=1

Step 3 — drop the modulus into two cases.

Case +1:

m+1/2=1-m/2 \Rightarrow 3m/2=1/2 \Rightarrow m=1/3

Case -1:

m+1/2=-1+m/2 \Rightarrow m/2=-3/2 \Rightarrow m=-3

Step 4 — sanity check. The two answers should be perpendicular to each other (the two 45° lines are at 90° to one another): 1/3\times \left( -3 \right)=-1. ✓

Answer: Option 3 — -3, \frac{1}{3}

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