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Question

If the curves, x2a+y2b=1 and x2c+y2d=1 intersect each other at an angle of 90°, then which of the following relations is true?


A

a+b=c+d

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B

a-b=c-d

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C

ab=c+da+b

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D

a-c=b+d

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Solution

The correct option is B

a-b=c-d


Explanation for the correct option

Step1: Solve for slopes of the curves

The given equation of the curves are

x2a+y2b=1_(i)

x2c+y2d=1_(ii)

Differentiate both sides of the equation i with respect to x.

ddxx2a+y2b=ddx12xa+2yb·dydx=0dydx=-bxay

Thus, slope of the curve, m1=dydx=-bxay.

Differentiate both sides of the equation ii with respect to x.

ddxx2c+y2d=ddx12xc+2yd·dydx=0dydx=-dxcy

Thus, slope of the curve, m2=dydx=-dxcy

Step 2: Solve for the required relation.

As the curves intersect with each other at an angle of 90°.

So, m1m2=-1

-bxay-dxcy=-1bdx2=-acy2x2=-acbdy2

Equation i-ii.

x2a+y2b-x2c+y2d=1-1x2a-x2c+y2b-y2d=01a-1cx2+1b-1dy2=0c-aac-acbdy2=-d-bbdy2a-cbdy2=b-dbdy2a-c=b-da-b=c-d

Therefore, the correct relation is a-b=c-d.

Hence, option(B) is the correct option.


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