Question

The D. E of the family of all circles in the first quadrant touching the coordinate axes.

• A. $[1+y_1^2{]}^3=r^2{y}_2^2$
• B. $[1+y_2{]}^3=r^2{y}_1^2$
• C. $[1+y_1^2{]}^2=r^2{y}_2^3$
• D. $[1+y_1{]}^3=r^2{y}_2^2$

Answer 1

The correct option is A $[1+y_1^2{]}^3=r^2{y}_2^2$

Equation is

$(x-r{)}^2+(y-r{)}^2=r^2$

$\Rightarrow 2(x-r)+2(y-r)y_1=0$

$(x-r)+(y-r)y_1=0$

$\Rightarrow 1+y_1^2+y_2(y-r)=0$

$\Rightarrow (y-r)=\frac{-(1+y_1^2)}{y_2}$

$r=y+\left(\frac{1+y_1^2}{y_2}\right)$

$(y-r{)}^2[1+y_1^2]=r^2$

$\Rightarrow \frac{(1+y_1^2{)}^3}{y_2^2}=r^2$

$\Rightarrow (1+y_1^2{)}^3=r^2{y}_2^2$