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)‌=0⇒‌‌‌‌(y−h)‌=−x⇒‌‌y−h=−‌ Now, substitute value of
y−h in equation of circle,
x2+‌=k2 ⇒‌‌x2(‌)2+x2=k2(‌)2 ⇒‌‌(x2−k2)(‌)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=0‌⇒‌‌x−h+y‌=0‌⇒‌‌h=x+y‌ Now, substitute the value of
h in the equation of circle
(x−x−‌)2+y2=(x+‌)2 ‌=x2−y2+(2xy)‌=0∴ Statement II is also true.