JEE Main 2019 — Parabola Question with Solution

Angle between the curves is the acute angle between the tangents at the point of intersection.

y = 10 $$-$$ x² (for curve 1)

and y = 2 + x² (for curve 2)

$$\therefore$$  10 $$-$$ x² = 2 + x²

$$\Rightarrow$$  2x² = 8

$$\Rightarrow$$  x² = 4

$$\Rightarrow$$  x = 2, $$-$$ 2

$$\therefore$$  points of intersection (2, 6) and ($$-$$ 2, 6)

$$\frac{dy}{dx}$$ for curve 1 = $$-$$ 2x

$$\therefore$$  Slope(m₁) of curve 1 is = $$-$$ 2(2) = $$-$$ 4

$$\frac{dy}{dx}$$ for curve 2 = 2x

$$\therefore$$  slope (m₂) of curve 2 = 2 $$\times$$ 2 = 4

$$\therefore$$  tan$$\theta$$ = $$\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$$

= $$\left|\frac{-4-4}{1+\left(-16\right)}\right|$$

= $$\frac{8}{15}$$