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Straight Lines and Pair of Straight Lines question

2019 · 10 Jan · Shift 1 · Q34
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Straight Lines and Pair of Straight Lines question

2019 · 10 Jan · Shift 1 · Q34

JEE MainMathematicsStraight Lines and Pair of Straight LinesMCQ+4 / −1
If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is :
  1. A
    (3, 4)
  2. B
    (2, 2)
  3. C
    (4, 4)
  4. D
    (4, 3)
View written solutionFree

Correct answer: B

  1. Find the intercepts of the line

The line is 3x+4y−24=0.3x + 4y - 24 = 0.3x+4y−24=0.

  • On the xxx-axis, y=0y=0y=0: 3x−24=0⇒x=8.3x-24=0 \Rightarrow x=8.3x−24=0⇒x=8. So, A=(8,0).A=(8,0).A=(8,0).

  • On the yyy-axis, x=0x=0x=0: 4y−24=0⇒y=6.4y-24=0 \Rightarrow y=6.4y−24=0⇒y=6. So, B=(0,6).B=(0,6).B=(0,6).

Also, the origin is O=(0,0).O=(0,0).O=(0,0).

So triangle OABOABOAB has vertices: O=(0,0),A=(8,0),B=(0,6).O=(0,0), \quad A=(8,0), \quad B=(0,6).O=(0,0),A=(8,0),B=(0,6).

  1. Identify the type of triangle

Since OAOAOA lies on the xxx-axis and OBOBOB lies on the yyy-axis, they are perpendicular. Hence, △OAB\triangle OAB△OAB is a right triangle with right angle at OOO.

Its side lengths are: OA=8,OB=6,AB=(8−0)2+(0−6)2=64+36=10.OA=8, \quad OB=6, \quad AB=\sqrt{(8-0)^2+(0-6)^2} = \sqrt{64+36}=10.OA=8,OB=6,AB=(8−0)2+(0−6)2​=64+36​=10.

  1. Use the incenter property for a right triangle

For a right triangle with legs along the coordinate axes and right angle at the origin, the incenter is at (r,r),(r,r),(r,r), where rrr is the inradius.

The inradius of a triangle is r=Areas,r = \frac{\text{Area}}{s},r=sArea​, where sss is the semiperimeter.

  • Area: Area=12⋅8⋅6=24.\text{Area} = \frac{1}{2}\cdot 8 \cdot 6 = 24.Area=21​⋅8⋅6=24.

  • Semiperimeter: s=8+6+102=12.s = \frac{8+6+10}{2}=12.s=28+6+10​=12.

Thus, r=2412=2.r=\frac{24}{12}=2.r=1224​=2.

Therefore the incentre is (2,2).(2,2).(2,2).

  1. Check options
  • A: (3,4)(3,4)(3,4) ❌
  • B: (2,2)(2,2)(2,2) ✅
  • C: (4,4)(4,4)(4,4) ❌
  • D: (4,3)(4,3)(4,3) ❌

Hence, the correct answer is Option B.

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