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 _____.
Answer (integer)
7
Solution
Let $P(3\cos \theta ,\,3\sin \theta )$<br><br>$Q( - 3\cos \theta ,\, - 3\sin \theta )$<br><br>$$\alpha = \left| {{{3\cos \theta + 3\sin \theta - 2} \over {\sqrt 2 }}} \right|$$<br><br>$$\beta = \left| {{{ - 3\cos \theta - 3\sin \theta - 2} \over {\sqrt 2 }}} \right|$$<br><br>$$\alpha \beta = \left| {{{{{\left( {3\cos \theta + 3\sin \theta } \right)}^2} - 4} \over 2}} \right|$$<br><br>$= \left| {{{5 + 9\sin 2\theta } \over 2}} \right|$<br><br>$\alpha {\beta _{\max }}$$= {{5 + 9} \over 2} = 7$ (when sin2$\theta$ = 1)
About this question
Subject: Mathematics · Chapter: Circles · Topic: Equation of a Circle
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