Let the normals at all the points on a given curve pass through a fixed point (a,b). If the curve passes through (3,−3) and 4,−2√2, and given that a−2√2b=3, then a2+b2+ab is equal to
A
7
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B
9
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C
6
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D
8
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Solution
The correct option is B9 Let the equation of normal is Y−y=−1m(X−x), where m=dydx
As it passes through (a,b) ⇒b−y=−1m(a−x)=−dxdy(a−x) ⇒(b−y)dy=(x−a)dx ⇒by−y22=x22−ax+c....(i)
It passes through (3,−3)&(4,−2√2) ∴−3b−92=92−3a+c ⇒3a−3b−c=9...(ii)
Also −2√2b−4=8−4a+c ⇒4a−2√2b−c=12...(iii)
Also a−2√2b=3…(iv) (given)
From (ii)−(iii) ⇒−a+(2√2−3)b=−3…(v)
From (iv)+(v) ⇒b=0,a=3 ∴a2+b2+ab=9