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MathematicsMultiple correct+4 / -2Question 12 of 54

JEE Advanced 2022 · Paper 1, Mathematics question 12

Let SS be the reflection of a point QQ with respect to the plane given by
r⃗=−(t+p)i^+tj^+(1+p)k^\vec r = -(t+p)\hat i + t\hat j + (1+p)\hat k
where t,pt, p are real parameters and i^,j^,k^\hat i, \hat j, \hat k are the unit vectors along the three positive coordinate axes. If the position vectors of QQ and SS are 10i^+15j^+20k^10\hat i + 15\hat j + 20\hat k and αi^+βj^+γk^\alpha\hat i + \beta\hat j + \gamma\hat k respectively, then which of the following is/are TRUE?

Choose one or more options
Answer and worked solution

Official answer: A, B, C

The plane. From x=−(t+p)x=-(t+p), y=ty=t, z=1+pz=1+p we get x+y+z=1x+y+z=1.

Reflect Q =(10,15,20)=(10,15,20). Here Q⋅(1,1,1)−1=44Q\cdot(1,1,1)-1=44, so
S=Q−2×443(1,1,1)=(−583,−433,−283).S=Q-\frac{2\times44}{3}(1,1,1)=\left(-\frac{58}3,-\frac{43}3,-\frac{28}3\right).

Check the options.

  • 3(α+β)=−1013(\alpha+\beta)=-101 ✓
  • 3(β+γ)=−713(\beta+\gamma)=-71 ✓
  • 3(γ+α)=−863(\gamma+\alpha)=-86 ✓
  • 3(α+β+γ)=−129≠−1213(\alpha+\beta+\gamma)=-129\ne-121 ✗
Topics:Equation of a planeAngles and distances
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