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Mathematics · properties of traingles

JEE Advanced 2020 — Paper 1 — Question 38

Let x,y\mathrm{x}, \mathrm{y} and z be positive real numbers. Suppose x,y\mathrm{x}, \mathrm{y} and z are the lengths of the sides of a triangle opposite to its angles X,Y\mathrm{X}, \mathrm{Y} and Z , respectively. If tan⁡x2+tan⁡z2=2yx+y+z\tan \frac{x}{2}+\tan \frac{z}{2}=\frac{2 y}{x+y+z} then which of the following statements is/are TRUE?

  1. Option A:

    2Y=X+Z2 Y=X+Z

  2. Option B:

    Y=X+ZY=X+Z

    Correct
  3. Option C:

    tan⁡x2=xy+z\tan \frac{x}{2}=\frac{x}{y+z}

    Correct
  4. Option D:

    x2+z2−y2=xzx^{2}+z^{2}-y^{2}=x z

Answer: B, C

Step-by-step solution

tan⁡X2+tan⁡Z2=2y2x+y+z\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2 y}{2 x+y+z}

⇒Δs(s−x)+Δs(s−z)=2y2s⇒Δ=(s−x)(s−z)\Rightarrow \frac{\Delta}{s(s-x)}+\frac{\Delta}{s(s-z)}=\frac{2 y}{2 s} \Rightarrow \Delta=(s-x)(s-z)

⇒s(s−y)=(s−x)(s−z)⇒z2+x2=y2\Rightarrow \mathrm{s}(\mathrm{s}-\mathrm{y})=(\mathrm{s}-\mathrm{x})(\mathrm{s}-\mathrm{z}) \Rightarrow \mathrm{z}^{2}+\mathrm{x}^{2}=\mathrm{y}^{2}

⇒\Rightarrow Triangle is right angled, y=π2\mathrm{y}=\frac{\pi}{2}

⇒Y=X+Z\Rightarrow \mathrm{Y}=\mathrm{X}+\mathrm{Z}

tan⁡x2=Δs(s−x)=12xz(x+y+z2)(y+z−x2)=xy+z\tan \frac{x}{2}=\frac{\Delta}{s(s-x)}=\frac{\frac{1}{2} x z}{\left(\frac{x+y+z}{2}\right)\left(\frac{y+z-x}{2}\right)}=\frac{x}{y+z}

Answer key and solution verified before publishing.

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Exam
JEE Advanced 2020
Paper
Paper 1
Subject
Mathematics
Chapter
properties of traingles
Topic
Sine,cosine,napier rules, half angle formula.area of triangle