Find the angle between the parabolas y2 = 4ax x2= 4by at their point of intersection other than - Maths - Application of Derivatives

Question

Find the angle between the parabolas y2 = 4ax &
x2= 4by at their point of intersection other than the
origin ?

Harleen Birdi · Accepted Answer

The given curves are y2 = 4ax (1) and x2 = 4by (2) Finding point of intersection of two curves: $x^{4} = 16 b^{2} y^{2} = 16 b^{2} (4 a x) = 64 a b^{2} x \Rightarrow x (x^{3} - 64 a b^{2}) = 0 \Rightarrow x = 0, 4 a^{\frac{1}{3}} b^{\frac{2}{3}}$ Now, x = 0 â‡’ y = 0 $and y^{2} = 4 a (4 a^{\frac{1}{3}} b^{\frac{2}{3}}) = 16 a^{\frac{4}{3}} b^{\frac{2}{3}} \Rightarrow y = 4 a^{\frac{2}{3}} b^{\frac{1}{3}}$ âˆ´ Points of intersection of (1) and (2) are (0, 0) and $(4 a^{\frac{1}{3}} b^{\frac{2}{3}}, 4 a^{\frac{2}{3}} b^{\frac{1}{3}})$ On differentiating (1) and (2), we get $\frac{d y}{d x} = \frac{2 a}{y} and \frac{d y}{d x} = \frac{x}{2 b}$ Angles between (1) and (2) at $(4 a^{\frac{1}{3}} b^{\frac{2}{3}}, 4 a^{\frac{2}{3}} b^{\frac{1}{3}})$ $m_{1} = \frac{d y}{d x} |_{(4 a^{\frac{1}{3}} b^{\frac{2}{3}}, 4 a^{\frac{2}{3}} b^{\frac{1}{3}})} = \frac{2 a}{4 a^{\frac{2}{3}} b^{\frac{1}{3}}} = \frac{a^{\frac{1}{3}}}{2 b^{\frac{1}{3}}} m_{2} = \frac{d y}{d x} |_{(4 a^{\frac{1}{3}} b^{\frac{2}{3}}, 4 a^{\frac{2}{3}} b^{\frac{1}{3}})} = \frac{4 a^{\frac{1}{3}} b^{\frac{2}{3}}}{4 b} = \frac{2 a^{\frac{1}{3}}}{b^{\frac{1}{3}}}$ Let Î¸ be the acute angle between (1) and (2) at $(4 a^{\frac{1}{3}} b^{\frac{2}{3}}, 4 a^{\frac{2}{3}} b^{\frac{1}{3}})$ , then $\tan θ = | \frac{m_{2} - m_{1}}{1 + m_{1} m_{2}} | = | \frac{\frac{2 a^{\frac{1}{3}}}{b^{\frac{1}{3}}} - \frac{a^{\frac{1}{3}}}{2 b^{\frac{1}{3}}}}{1 + \frac{2 a^{\frac{1}{3}}}{b^{\frac{1}{3}}} \frac{a^{\frac{1}{3}}}{2 b^{\frac{1}{3}}}} | = \frac{3 a^{\frac{1}{3}} b^{\frac{1}{3}}}{2 (a^{\frac{2}{3}} + b^{\frac{2}{3}})} \Rightarrow θ = \tan^{- 1} (\frac{3 a^{\frac{1}{3}} b^{\frac{1}{3}}}{2 (a^{\frac{2}{3}} + b^{\frac{2}{3}})})$

Find the angle between the parabolas y2 = 4ax & x2= 4by at their point of intersection other than the origin ?

Find the angle between the parabolas y² = 4ax & x²= 4by at their point of intersection other than the origin ?