Answer to Question #273123 in Algebra for Holi

Question

Consider the equation and the relation “(x, y) R (0, 2)”, where R is read as “has distance 1 of”. For example, “(0, 3) R (0, 2)”, that is, “(0, 3) has distance 1 of (0, 2)”. This relation can also be read as “the point (x, y) is on the circle of radius 1 with center (0, 2)”. In other words: “(x, y) satisfies this equation , if and only if, (x, y) R (0, 2)”.

Does this equation determine a relation between x and y?

Can the variable x can be seen as a function of y, like x=g(y)?

Can the variable y be expressed as a function of x, like y= h(x)?

If these are possible, then what will be the domains for these two functions?

What are the graphs of these two functions?

Are there points of the coordinate axes that relate to (0, 2) by means of R?

Answer 1

(i) Yes, this determine the relation x and y are chosen in such a way that these form a circle 1 unit radius.

therefore to the equation determine a relation between x and y

Hence we can say that x and y repeated each other i.e

$(x-a)^2+(y-b)^2=1$, where (a,b) is the centre of the circle

(ii) y/ (a,b) is (0,2)

$then\ x^2+(y-2)^2=1$

$x=g(y)=\sqrt{1-(y-2)^2}$

this is relation which shows the dependence of x and y

$y=h(x)=\sqrt{1-x^2}+2$

then the domain of g(y) is is [1,3] as y will take those value then domain of

$h(x)=[-1,1]$

(iii) the graphs of these two are half circle with center

$(0,2)$

(iv) All of the points on the circle that have distance 1 from point (0,2)

point (0,2) is related R as it will represent circle of 1 unit radius