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Question

The sides BC,CA, and AB of a ABCare of lengths a, b and c respectively. If D is the mid-point of BCand AD is perpendicular to AC,then the value of cosAcosC is


A

2(c2a2)3ac

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B

3(c2a2)2ac

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C

1

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D

None of these

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Solution

The correct option is A

2(c2a2)3ac


Explanation for the correct option:

Step 1: Interpret the given data

ABC is a triangle

BC,CA, and ABare of length a,b,and c, respectively.

D is the mid-point of BC

AD is perpendicular to AC

Step 2: Apply cosine rule

The cosine rule states that, in a triangle with sides a, b and c with opposite angles A, B and C respectively.
c2=a2+b2-2abcosC

This applies for any side of the triangle. It is an extension of the Pythagoras theorem.

From, right ACD

cosC=ba2=2ba(1)

By cosine rule

c2=a2+b2-2abcosCcosC=a2+b2-c22ab(2)2ba=a2+b2-c22abb2=a2-c23(3)

Also, by cosine rule

cosA=b2+c2-a22bc

Then,

cosAcosC=b2+c2-a22bc×2ba=b2+c2-a2ac=a2-c23+c2-a2acfromeqn3=a2-c2+3c2-3a23accosAcosC=2c2-2a23ac

Hence, option A is correct.


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