The supply price relation is given by
\({ x }^{ 2 }=5p-15\) 
= \(5(p-3)\) 
\(\Rightarrow { X }^{ 2 }aP\)  where X = x and \(P=p-3\) 
\(\\ \therefore \) the supply curve is a parabola whose vertex is \(\left( X=0,\ P=0 \right) \) 
i.e., The supply curve is a parabola whose vertex is (0, 3)

The length of the tangent to the circle \({ x }^{ 2 }+{ y }^{ 2 }+2gx+2fy+c=0\) from a point \(\left( { x }_{ 1 },{ y }_{ 1 } \right) \) is \(\sqrt { { x }_{ 1 }^{ 2 }+{ y }_{ 1 }^{ 2 }+2{ gx }_{ 1 }+2{ fy }_{ 1 }+c } \) 
Length of the tangent
\(\\ =\sqrt { { x }_{ 1 }^{ 2 }+{ y }_{ 1 }^{ 2 }+8{ x }_{ 1 }+4{ y }_{ 1 }+8 } \)
\(=\sqrt { { 2 }^{ 2 }+{ 3 }^{ 2 }+8(2)+4(3)+8 } \)
\( \left[ Here({ x }_{ 1 }{ y }_{ 1 })=(2,3) \right] \)
\(=\sqrt { 49 } \)
Length of the tangent = 7 units

The circle \({ x }^{ 2 }+{ y }^{ 2 }+ax+by=0\) passing through (1, 2) and (1, 1)
\(\therefore \) We have 1 + 4 + a + 2b = 0 and  1 + 1 + a + b = 0
\(\Rightarrow a+2b=-5\) (1)
and a + b = –2  (2)
Solving (1) and (2),we get a = 1,b = -3

Centre C(–1, 3)
It lies on ax + 2y + 2 = 0
-a + 6 + 2 = 0
a = 8

Equation of a circle when the end points of the diameter are given is
\(\left( x-{ x }_{ 1 } \right) \left( x-{ x }_{ 2 } \right) +\left( y-{ y }_{ 1 } \right) \left( y-{ y }_{ 2 } \right) =0\) 
Here \(\left( { x }_{ 1 },{ y }_{ 1 } \right) =\left( 2,4 \right) \) and
\(\left( { x }_{ 2 },{ y }_{ 2 } \right) =\left( 3,-2 \right) \)
\(\therefore \left( x-2 \right) \left( x-3 \right) +\left( y-4 \right) \left( y+2 \right) =0\)
\({ x }^{ 2 }+{ y }^{ 2 }-5x-2y-2=0\)