TS EAMCET 19-July-2022 Shift 2 Solved Paper

Statement I : If the centre of circle lies on Y-axis and radius k, then equation of circle is (x)2+(y−h)2=k2, where (0,h) is the centre of circle.
‌2x+2(y−h)‌

dy

dx

=0
⇒‌‌‌‌(y−h)‌

dy

dx

=−x
⇒‌‌y−h=−‌

x

dy∕dx

Now, substitute value of y−h in equation of circle,
x2+‌

x2

(‌ dy dx )2

=k2
⇒‌‌x2(‌

dy

dx

)2+x2=k2(‌

dy

dx

)2
⇒‌‌(x2−k2)(‌

dy

dx

)2+x2=0
∴ Statement I is true.
Statement II : If the centre of circle lies on X-axis and circle passes through origin is (x−h)2+y2=h2, where centre of circle is (h,0) and radius is h unit.
On differentiate both sides w.r.t. x, we get
‌2(x−h)+2y‌

dy

dx

=0
‌⇒‌‌x−h+y‌

dy

dx

=0
‌⇒‌‌h=x+y‌

dy

dx

Now, substitute the value of h in the equation of circle
(x−x−‌

ydy

dx

)2+y2=(x+‌

ydy

dx

)2
‌=x2−y2+(2xy)‌

dy

dx

=0
∴ Statement II is also true.