KVPY Sample Paper KVPY Stream-SX Model Paper-17

If \[{{r}_{1}}\] and \[{{r}_{2}}\] are the distances from the origin of points on the curve \[10(z\bar{z})-3i\{{{z}^{2}}-{{(\bar{z})}^{2}}\}-6=0\]. Which are at maximum and minimum distance from the origin then the value of \[{{r}_{1}}+{{r}_{2}}\] is equal to

A) 4

B) 3

C) 2

D) none of these

Correct Answer: B

Solution :

The equation is \[10(z\bar{z})-3i\{{{z}^{2}}-{{(\bar{z})}^{2}}\}-6=0\]
Or \[10({{x}^{2}}+{{y}^{2}})-3i(2x)(2iy)-6=0\]
\[\Rightarrow \]\[5({{x}^{2}}+{{y}^{2}})+6xy-8=0\]	? (1)
Let \[(r\cos \theta ,r\sin \theta )\]be a point on (1), then \[5{{r}^{2}}+6{{r}^{2}}\sin \theta \cos \theta -8=0\]

\[\Rightarrow \]\[{{r}^{2}}=\frac{8}{5+3\sin 2\theta }\]

Clearly, \[1\le {{r}^{2}}\le 4\Rightarrow 1\le \left| r \right|\le 2\]

\[\therefore \]\[{{r}_{1}}={{\left| r \right|}_{\max }}=2\] and \[{{r}_{2}}={{\left| r \right|}_{\max }}=1\]

\[\Rightarrow \]\[{{r}_{1}}+{{r}_{2}}=3\]