Question

If A = {(x, y) : x² + y² = 25} and B = {(x, y) : x² + 9y² = 144}. Then find the common elements between set A and B i.e. A ∩ B.

Answer 1

Intersection - Sets

A well-defined collection of objects is called set. The object in a set are called its member or elements. Whose elements are fixed and cannot vary.

• lt means the set doesn't change from person to person.

• Like for example, the set of natural numbers up to 5 will remain the same as {1,2,3,4,5}.

Hint: Assume $(x, y)$ is in both sets $A$ and $B$. Hence using the definition of sets form two equations in $x$ and $y$. Solve for $x$ and $y$. The number of solutions of the system is the number of points in $A \cap B$.

Step-by-step solution:

We've been given two sets A and B. With this information, we've been asked to find common elements between set A and B. i.e.,

• A = {(x, y) : x² + y² = 25}
• B = {(x, y) : x² + 9y² = 144}
• A ∩ B contains?

First of all we'll solve two given equations out of which we will get the value of $x$ and $y$ and by that we will know how many points $A$ and $B$ contains.

Given curves are;

$x^2 + y^2 = 25 \qquad .... .(1)$

$x^2 + 9y^2 = 144 \qquad .... .(2)$

Subtracting equation (1) from equation (2), we obtain:

$\implies (x^2 + 9y^2) - (x^2 + y^2) = 144 - 25$

$\implies x^2 + 9y^2 - x^2 - y^2 = 144 - 25$

$\implies x^2 - x^2 + 9y^2 - y^2 = 144 - 25$

$\implies 0 + 9y^2 - y^2 = 144 - 25$

$\implies 8y^2 = 199$

$\implies y^2 = \dfrac{199}{8}$

$\implies y = \pm\sqrt{\dfrac{199}{8}}$

$\implies y = \pm\dfrac{\sqrt{199}}{2\sqrt{2}}$

Now substituting the value of $y^2$ in equation (1), we get:

$\implies x^2 + \dfrac{199}{8} = 25$

$\implies x^2 = 25 - \dfrac{199}{8}$

$\implies x^2 = \dfrac{200 - 199}{8}$

$\implies x^2 = \dfrac{81}{8}$

$\implies x = \pm\sqrt{\dfrac{81}{8}}$

$\implies x = \pm\dfrac{9}{2\sqrt{2}}$

So equation (1) and equation (2) intersect in four points:

$\boxed{\bigg(\dfrac{9}{2\sqrt{2}}, \dfrac{\sqrt{199}}{2\sqrt{2}}\bigg), \bigg(-\dfrac{9}{2\sqrt{2}}, -\dfrac{\sqrt{199}}{2\sqrt{2}}\bigg), \bigg(-\dfrac{9}{2\sqrt{2}}, \dfrac{\sqrt{199}}{2\sqrt{2}}\bigg), \bigg(\dfrac{9}{2\sqrt{2}}, -\dfrac{\sqrt{199}}{2\sqrt{2}}\bigg)}$

Hence, it is clear that A ∩ B consists of four points.

Answer 2

Answer:

Step-by-step explanation:Given : A = (x,y)(x²+y²=25) and B=(x,y)(x²+9y²=144)

To find : A ∩ B

Solution:

x²  + y ²  = 25

x²  + 9y²  = 144

=> 8y²  = 119

=> y² = 119/8

=> y  = ± √ 119/ 2√2

Substitute y² = 119/8  

in     x²  + y ²  = 25

=> x²  + 119/8  = 25

=> x² = 81/8

=> x  = ± 9/2√2

4 Possible points area

(   9/2√2 ,  √ 119/ 2√2 )   ,

(   9/2√2 ,  -√ 119/ 2√2 )  

(   -9/2√2 ,  √ 119/ 2√2 )  

(   -9/2√2 ,  -√ 119/ 2√2 )  

A ∩ B  = 4

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