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Let PQ be a diameter of the circle x2 y2 9 If alpha and... | null (Mathematics) | DivineJEE

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Mathematics

Exam

Mathematics

Let PQ be a diameter of the circle x² + y² = 9. If

\alpha

and

\beta

are the lengths of the perpendiculars from P and Q on the straight line,
x + y = 2 respectively, then the maximum value of

\alpha\beta

is _____.

Correct Answer:

7

Explanation

Let

P(3\cos \theta ,\,3\sin \theta )

Q( - 3\cos \theta ,\, - 3\sin \theta )

\alpha = \left| {{{3\cos \theta + 3\sin \theta - 2} \over {\sqrt 2 }}} \right|

\beta = \left| {{{ - 3\cos \theta - 3\sin \theta - 2} \over {\sqrt 2 }}} \right|

\alpha \beta = \left| {{{{{\left( {3\cos \theta + 3\sin \theta } \right)}^2} - 4} \over 2}} \right|

= \left| {{{5 + 9\sin 2\theta } \over 2}} \right|

\alpha {\beta _{\max }}

= {{5 + 9} \over 2} = 7

(\text{when sin2} \theta = 1)

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