Tangents and normals

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$The angle between the parabolas y^{2} = 4 a x and x^{2} = 4 a y at their point of intersection other than the origin.$

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$\begin{array}{l} Pt of intersection of y^{2} = 4 a x & x^{2} = 4 a y \\ are (0, 0) (4 a, 4 a) \\ 2 y \frac{d y}{d x} = 4 a \Rightarrow \frac{d y}{d x} = \frac{2 a}{y} \\ m_{1} = {(\frac{d y}{d x})}_{(4 a, 4 a)} = \frac{2 a}{4 a} = \frac{1}{2} \\ m_{2} =_{(4 a, 4 a)} = \frac{2 x}{4 a} = \frac{8 a}{4 a} = 2 \end{array}$
$|\tan θ = \frac{2 - \frac{1}{2}}{1 + 1}| = \frac{3}{4}$
$\tan θ = \frac{3}{4} \Rightarrow θ = \tan^{- 1} (\frac{3}{4})$

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Tangents and normals

Remember concepts with our Masterclasses.

The angle between the parabolas y2=4ax and x2=4ay at their point of intersection other than the origin.

Pt of intersection of y2=4ax&x2=4ay are (0,0) (4a,4a)2ydydx=4a⇒dydx=2aym1=dydx(4a,4a)=2a4a=12m2=(4a,4a)=2x4a=8a4a=2 tanθ = 2−121+1 = 34tanθ = 34 ⇒θ = tan−1(34)

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$The angle between the parabolas y^{2} = 4 a x and x^{2} = 4 a y at their point of intersection other than the origin.$