Question

The acute angles between the curves y = |x² – 1| and

y = |x² – 3| at their points of intersection is –

• A. p/4
• B. tan^–1 (4 Ö2/7)
• C. tan^–1 (4 Ö7)
• D. None of these

Answer 1

Answer: (B)

tan^–1 (4 Ö2/7)

Answer 2

Clearly when x² = 2, y = 1 gives the point of intersection. The given equations represent four parabolas y = ± (x² – 1) and y = ± (x² – 3) which can be traced. The curves y = x² – 1 and y = –(x² – 3) intersect when 1 < x² < 3 or 1 < x < Ö3 or – Ö3 < x < –1

The points of intersections are (± Ö2, 1). At (Ö2, 1)

m₁ = 2x = 2Ö2, m₂ = – 2x = –2Ö2

\ tan q = $\left| \frac{4\sqrt{2}}{1 - 8} \right|$ = $\frac{4\sqrt{2}}{7}$, \ q = tan^–1 $\left( \frac{4\sqrt{2}}{7} \right)$

In a similar manner the acute angles at the other points in same as found above.