Math, asked by nitnyadav, 10 days ago

If A = [(x, y) : x2 + y2 = 25] and B = [(x, y) : x2 + 9y2 = 144], then A ∩ B contains _______ points.

Answers

Answered by mathdude500
12

Answer:

\qquad\boxed{ \sf{ \:\bf\  \: A \cap B \: contains \: 4 \: points.\: }} \\  \\

Step-by-step explanation:

Given that,

\sf \: A =  \{(x,y) :  {x}^{2}  +  {y}^{2}  = 25 \:  \} \\  \\

and

\sf \: B =  \{(x,y) :  {x}^{2}  + 9{y}^{2}  = 144 \:  \} \\  \\

Now,

\sf \:  {x}^{2} +  {y}^{2}  = 25 \: represents \: equation \: of \: circle. \\  \\

\sf\implies \: A \: consist \: of \: all \: those \: points \: which \: lies \: on \: circle. \\  \\

Further,

\sf \:  {x}^{2} +  9{y}^{2}  = 144 \: represents \: equation \: of \: ellipse. \\  \\

\sf\implies \: B \: consist \: of \: all \: those \: points \: which \: lies \: on \: ellipse. \\  \\

So,

\sf\implies A \cap B \: is \: set \: of \: point \: of \: intersection \: of \: circle \: and \: ellipse. \\  \\

So, Let's evaluate the point of intersection of

\sf \:  {x}^{2} +  {y}^{2}  = 25 \:  \:  -  -  - (1) \\  \\

and

\sf \:  {x}^{2} +  9{y}^{2}  = 144 \:  \:  -  -  - (2) \\  \\

On Subtracting equation (1) from (2), we get

\sf \:  {8y}^{2} = 119 \\  \\

\sf \:  {y}^{2} = \dfrac{119}{8}  \\  \\

\sf\implies \sf \:  y =  \pm \: \dfrac{ \sqrt{119} }{2 \sqrt{2} }  \\  \\

On substituting the value of y^2 in equation (1), we get

\sf \:  {x}^{2}  + \dfrac{119}{8}  = 25 \\  \\

\sf \:  {x}^{2}   = 25 -  \dfrac{119}{8}  \\  \\

\sf \:  {x}^{2}   =  \dfrac{200 - 119}{8}  \\  \\

\sf \:  {x}^{2}   =  \dfrac{81}{8}  \\  \\

\sf\implies  \: x \:  =  \:  \pm \: \dfrac{9}{2 \sqrt{2} }  \\  \\

Hence, point of intersection are

\qquad\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf  \dfrac{9}{2 \sqrt{2} }  & \sf \dfrac{ \sqrt{119} }{2 \sqrt{2} }  \\ \\ \sf  \dfrac{9}{2 \sqrt{2} }  & \sf  -\dfrac{ \sqrt{119} }{2 \sqrt{2} }   \\ \\ \sf  - \dfrac{9}{2 \sqrt{2} }  & \sf \dfrac{ \sqrt{119} }{2 \sqrt{2} }  \\ \\ \sf  - \dfrac{9}{2 \sqrt{2} }  & \sf -  \dfrac{ \sqrt{119} }{2 \sqrt{2} }  \end{array}} \\ \end{gathered} \\  \\

Hence,

\bf\implies  \: A \cap B \: contains \: 4 \: points. \\  \\

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Answered by Itsanuragjaiswal
12

\large{\sf{Question:-}}

If A = [(x, y) : x2 + y2 = 25] and B = [(x, y) : x2 + 9y2 = 144], then A ∩ B contains _______ points.

Solutions with given :

Clearly, A is the set of all points on the circle x² +y² =25 and B is the set of all points on the ellipse x² +9y² =144. These two intersect at four points P,Q,R and S.

Hence, A∩B contains four points.

Answer:-

\begin{gathered}\qquad\qquad\boxed{\bf{A∩B \:contains \:four \:points}}\end{gathered}

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