Verified Solution • 2025–26 Session moderate

QJEEM20Mab731: JEE Main 2020 Mathematics - Class 11 JEEM20Mab731 Solution: Let PQ be a diameter of the circle x 2 + y 2 = 9. If and are…

Let PQ be a diameter of the circle x2 + y2 = 9. If α\alpha and β\beta are the lengths of the perpendiculars from P and Q on the straight line,
x + y = 2 respectively, then the maximum value of αβ\alpha\beta is _____.

Step-by-Step Verified Solution

Let P(3cos⁡θ, 3sin⁡θ)P(3\cos \theta ,\,3\sin \theta )

Q(−3cos⁡θ, −3sin⁡θ)Q( - 3\cos \theta ,\, - 3\sin \theta )

α=∣3cos⁡θ+3sin⁡θ−22∣\alpha = \left| {{{3\cos \theta + 3\sin \theta - 2} \over {\sqrt 2 }}} \right|

β=∣−3cos⁡θ−3sin⁡θ−22∣\beta = \left| {{{ - 3\cos \theta - 3\sin \theta - 2} \over {\sqrt 2 }}} \right|

αβ=∣(3cos⁡θ+3sin⁡θ)2−42∣\alpha \beta = \left| {{{{{\left( {3\cos \theta + 3\sin \theta } \right)}^2} - 4} \over 2}} \right|

=∣5+9sin⁡2θ2∣= \left| {{{5 + 9\sin 2\theta } \over 2}} \right|

αβmax⁡\alpha {\beta _{\max }} =5+92=7= {{5 + 9} \over 2} = 7 (when sin2 θ\theta = 1)

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring