Condition that 2 curves y²=4ax, xy=c² cut orthogonally is

Correct Answer is : c4=32a4c^{4}=32a^{4}c4=32a4

Correct Answer is : c4=32a4c^{4}=32a^{4}c4=32a4

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AP EAPCET 04-Jul-2022 Shift 2 Paper

Section: Mathematics

Question : 67 of 160

Marks: +1, -0

Condition that 2 curves

y^{2}=4ax, xy=c^{2}

cut orthogonally is

Solution: 👈: Video Solution

(d) Given,

y^{2}=4ax

and

xy=i^{2}

cut orthogonally.

Let they intersect at

(x_{1},y_{1})

\therefore \;\; y^{2}=4ax \Rightarrow 2y\;\frac{dy}{dx}=4a

\Rightarrow \;\frac{dy}{dx}=\;\frac{2a}{y}

\therefore \;\;\frac{dy}{dx}(x_{1},y_{1})=\;\frac{2a}{y_{1}}........(i)

and

\;\; xy=c^{2}

\;\Rightarrow \;\; y+x\;\frac{dy}{dx}=0

\;\Rightarrow \;\; \;\frac{dy}{dx}(x_{1},y_{1})=-\;\frac{y_{1}}{x_{1}}......(ii)

From Eqs (i) and (ii), we get

\;\; \;\frac{2a}{y_{1}}\times-\;\frac{y_{1}}{x_{1}}=-1

\Rightarrow \;\; x_{1}=2a

From

y^{2}=4ax

\;\Rightarrow \;\; y_{1}=\sqrt{4a\cdot2a}=2\sqrt{2}a \Rightarrow x_{1}y_{1}=c^{2}

\;\Rightarrow \;\; 2a(2\sqrt{2}a)=c^{2} \Rightarrow \;\frac{c^{2}}{a^{2}}=4\sqrt{2}

\;\Rightarrow \;\; c^{4}=32a^{4}

\;

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