(a) (i) Given equation of circle is
(x - 2)2 + (y + 3)2 = 9
⇒ (x - 2)2 + y(-(-3))2 = 32
\(\therefore\) Centre of circle is (2, -3) and radius of the circle is 3.(But comparing the standard form of circle whose centre is (h, k) and radius is r; i.e; (x - h)2 + (y - k)2 = r2)
(ii) Given equation of circle is
2x2 + 2y2 - 6x + 10y = 1
⇒ x2 + y2 - 3x + 5y = 1/2 (Dividing both sides by 2)
Compare with x2 + y2 + 2fx + 2gy + c = 0
we get 2f = -3 ⇒ f = -3/2
2g = 5 ⇒ g = 5/2
& c = -1/2
\(\therefore\) Centre of the circle = (-f, -g) = (3/2, -5/2)
and radius of the circle is r = \(\sqrt{f^2+g^2-c}\)
⇒ r = \(\sqrt{(\frac{-3}2)^2+(5/2)^2-(-1/2)}\)
= \(\sqrt{\frac94+\frac{25}4+\frac12}\)
= \(\sqrt\frac{9+25+2}4\) = \(\sqrt\frac{36}4=\sqrt9\) = 3
\(\therefore\) Radius of the circle is r = 3
& centre of the circle is(-3/2, 5/2)
(iii) Given equation of circle is
x2 + y2 + 8x - 2y + 13 = 0
⇒ x2 + 8x + 16 - 16 + y2 - 2y + 1 - 1 + 13 = 0
⇒ (x + 4)2 + (y - 1)2 + 13 - 17 = 0
⇒ (x - (-4))2 + (y - 1)2 = 22
\(\therefore\) Centre of the circle is(-4, 1)
and radius of the circle is r = 2
(b)(i) Polar equationo of the curve is
\(\frac2r=1+cos\theta\)----(1)
\(\because\) x = r cos θ and y = r sin θ
\(\therefore\) cos θ = x/r
Then equation(1) converts into
2/r = 1 + x/r
⇒ 2 = x + r
⇒ 2 - x = r
⇒ r2 = (2 - x)2
⇒ x2 + y2 = x2 - 4x + 4 (\(\because\) r2 = x2 + y2)
⇒ y2 = -4x + 4 which is a cartesian equation of the given curve.
(ii) Given cartesian equation of the curve is
x2 + y2 = 9
⇒ r2 = 9 (\(\because\) x2 + y2 = r2)
⇒ r = 3
(c) (i) Equation of line which passes through two points (-2, 3) & (2, -5) is
\(y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)\)
⇒ y - 3 = \(\frac{-5-3}{2-(-2)}(x - (-2))\)
⇒ 2x + y + 1 = 0 which is equation of required line.
(ii) Given equation of curve is
x2 + y2 + 2x + 4y - 3 = 0----(1)
By diferentiating equation (1) w.r.t. x
we get
2x + 2y dy/dx + 2 + 4 dy/dx = 0
⇒ (2y + 4)dy/dx = -(2x + 2)
⇒ dy/dx = \(\frac{-(2x+2)}{2y+4} = \frac{-(x+1)}{y+2}\)
\(\therefore\) slope of tangent to curve (1) at point (1, -4) is
m = (dy/dx)(1, -4) = \(\frac{-(1+1)}{-(4+2)}=\frac{-2}{-2}=1\)
\(\therefore\) Equation of tangent to curve (1) at point (1, -4) is y-(-4) = m(x - 1)
⇒ y + 4 = 1(x - 1)
⇒ x - y = 5
which is equation of tangent to given curve at point(1, - 4)