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NCERT Class 12 · Ch 6 Miscellaneous Exercise · Q5

NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Miscellaneous Exercise Question 5

Find the maximum area of an isosceles triangle inscribed in x²/a²+y²/b²=1, vertex at an end of the major axis.
Ellipse representative a=2, b=1: optimal isosceles triangle
Ellipse representative a=2, b=1: optimal isosceles triangle

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: 3√3 ab/4.
  1. Take a≥b>0 and vertex (a,0); the symmetric base endpoints are (u,±b√(1−u²/a²)), −a<u<a. If the major axis is vertical interchange the axes.
  2. Area A(u)=b(a−u)√(1−u²/a²). Set v=u/a; A²=a²b²(1−v)³(1+v).
  3. Differentiate the factor: d[(1−v)³(1+v)]/dv=−2(1−v)²(1+2v). The unique interior maximum is v=−1/2.
  4. Thus u=−a/2, base length √3b and altitude 3a/2, giving Amax=3√3ab/4.

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