P and Q are the ends of a diameter of the circle x2+y2=a2(a>‌
1
√2
)â‹…s and t are the lengths of the perpendiculars drawn from P and Q onto the line x+y=1 respectively. When the product st is maximum, the greater value among s,t is
Given, equation of circle is x2+y2=a2‌‌‌⋅⋅⋅⋅⋅⋅⋅(i) Also P and Q are end points of diameter. Let P≡(a‌cos‌θ,asin‌θ) Q≡(−a‌cos‌θ,−asin‌θ) According to the question, s‌=‌
|a‌cos‌θ+asin‌θ−1|
√2
‌=‌
|a(cos‌θ+sin‌θ)−1|
√2
‌ and ‌t‌=‌
|−a(cos‌θ+sin‌θ)−1|
√2
=1 ‌ Now, st ‌‌=‌
|1−a2(cos‌θ+sin‌θ)2|
2
‌=‌
|1−a2(1+sin‌2θ)|
2
So, ( st ) will be maximum, if ( 1+sin‌2θ ) minimum, ‌∵(sin‌2θ)min=−1(‌ i.e. ‌sin‌2θ≥−1) ‌⇒1+sin‌2θ≥0∴(1+sin‌2θ)min=0 ‌∴(a2(1+sin‌2θ))min=0‌ and hence, ‌st=‌