(i) Let the straight line be
y=mx+c1Since it is tangent to the circle x2+y2=a2
So, a=|0+0−c1|√1+m2
c1=a√(1+m2) or, c1=−a√(1+m2)
Hence, the tangents are
y=mx+a√(1+m2) and y=mx−a√(1+m2)
(ii) Let the straight line be y=m1x+c2
m1=−1m
Since it is tangent to the circle x2+y2=a2
So, a=|0+0−c2|√1+1m2
c2=a(√1+m2)m or c2=−a(√1+m2)m
So, the equation of tangent are
y=−1mx+a(√1+m2)m and y=−1mx−a(√1+m2)m
(iii) Let the straight line be y=mx+c
The point passes through (b,0)
So, 0=bm+c
c=−bm
Since it is the tangent to the circle x2+y2=a2
So,a=|0+0+bm|1+m2
Solving we get the tangent as
y=√a2b2−a2x−bm and y=−√a2b2−a2x−bm
(iv) Let the straight line be y=mx+c
It cuts x-axis at (−cm,0) and y-axis at (0,c).
Area of triangle is a2 which is also equal to 12.cm.c.
⇒c22m=a2
Since it is the tangent to the circle x2+y2=a2
So, a=|0+0−c|√1+m2
On solving we get,
m=1 and c=√2a
So, the tangent is, y=x+√2a.