AP EAMCET · Maths · Application of Derivatives

Condition that $2$ curves $y^{2} = 4 a x, x y = c^{2}$ cut orthogonally is

A $c^{2} = 16 a^{2}$
B $c^{2} = 32 a^{2}$
C $c^{4} = 16 a^{4}$
D $c^{4} = 32 a^{4}$

Verified Solution

Answer & Solution

Correct Answer

(D) $c^{4} = 32 a^{4}$

Step-by-step Solution

Detailed explanation

Given y2=4ax, xy=c2 For y2=4ax, 2ydydx=4a i.e. dydxx1,y1=2ay1 For xy=c2, xdydx+y=0 i.e. dydxx1,y1=-y1x1 For orthogonal intersection of both curves 2ay1×-y1x1=-1 i.e. x1=2a Also, y12=4ax1 ⇒y12=8a2 and x1y12=c22 ⇒c4=4a2×8a2=32a4

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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

Step-by-step Solution

Condition that $2$ curves $y^{2} = 4 a x, x y = c^{2}$ cut orthogonally is