y = k represents …………… A) parallel to X – axis  B) parallel to Y – axis  C) parallel to X, Y axes  D) (0, 0)

Correct option is B) parallel to Y – axis

Correct option is (B) Y-axis Equation x = 0 represents Y-axis.

Correct option is (A) X-axis Equation y = 0 represents X-axis.

Correct option is (B) parallel to Y–axis Equation of Y-axis is x = 0. \(\therefore\) Equation x = 0 represents Y-axis. \(\therefore\) Equation x = k represents line parallel to Y-axis.

Equation of the line passing though (2,3) which is (i) parallel to x-axis is y = 3 and (ii) parallel to y-axis is x = 2.

With the given notation, we have `x=X+4,y=Y+5` or `X=x-4,Y=y-5` Therefore, the required equation is `(x-4)^2+(y-5)^2=36` or `x^2+y^2-8x-10y+5=0` which is the eqaution referred to the original axes.

We have `x=Xcostheta-Ysin theta` `=X cos theta 45^@-Ysin45^@=(X)/(sqrt2)+(Y)/(sqrt2)` `y=X sintheta+Ycostheta` `=X sintheta 45^@-Ycos45^@=(X)/(sqrt2)+(Y)/(sqrt2)` Hence, the equation `3x^2+2xy+3y^2=10` transforms to `3((X)/(sqrt2)-(Y)/(sqrt2))^2+2((X)/(sqrt2)-(Y)/(sqrt2))((X)/(sqrt2)+(Y)/(sqrt2))+3((X)/(sqrt2)+(Y)/(sqrt2))^2=10` or `2X^2+Y^2=5`