Chapter 10 - Conic Sections

Radius $r = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = 5$.
Area = $\pi r^2 = 25\pi$.

Since the circle touches both axes and passes through (3, 6) (1st quadrant), its centre is $(r, r)$ and radius is $r$.
Equation: $(x-r)^2 + (y-r)^2 = r^2$.
Passes through (3, 6): $(3-r)^2 + (6-r)^2 = r^2 \Rightarrow r^2 - 18r + 45 = 0 \Rightarrow r=3, 15$.
For $r=3$: $(x-3)^2 + (y-3)^2 = 9 \Rightarrow x^2 + y^2 - 6x - 6y + 9 = 0$.

Centre on y-axis $(0, k)$. Passes through origin $\Rightarrow r=|k|$.
Equation: $x^2 + (y-k)^2 = k^2 \Rightarrow x^2 + y^2 - 2ky = 0$.
Passes through (2, 3): $4 + 9 - 6k = 0 \Rightarrow k = 13/6$.
Equation: $x^2 + y^2 - 13/3 y = 0 \Rightarrow 3x^2 + 3y^2 - 13y = 0$.
None of the given options match.

Centroid coincides with the centre (origin). Radius $r = \frac{2}{3} \times \text{median} = \frac{2}{3}(3a) = 2a$.
Equation: $x^2 + y^2 = (2a)^2 = 4a^2$.

Focus $(0, -3)$, Directrix $y=3$. Vertex is $(0,0)$. Parabola opens downwards.
$a = 3$. Equation: $x^2 = -4ay = -12y$.

Substitute (3, 2) in $y^2 = 4ax$: $4 = 4a(3) \Rightarrow 4a = 4/3$.
Length of latus rectum is $4a = 4/3$.

Vertex $(-3, 0)$, Directrix $x = -5$. Distance $a = 2$.
Opens right. Equation: $(y-0)^2 = 4(2)(x - (-3)) \Rightarrow y^2 = 8(x+3)$.

Using $SP = e PM$: $\sqrt{(x-1)^2 + (y+1)^2} = \frac{1}{2} \frac{|x-y-3|}{\sqrt{2}}$.
Squaring: $(x-1)^2 + (y+1)^2 = \frac{1}{8} (x-y-3)^2$.
$8(x^2-2x+1+y^2+2y+1) = x^2+y^2+9-2xy-6x+6y$.
$7x^2 + 7y^2 + 2xy - 10x + 10y + 7 = 0$.

Equation: $\frac{x^2}{4} + \frac{y^2}{12} = 1$. $a^2=4, b^2=12$. $b > a$.
Latus Rectum = $\frac{2a^2}{b} = \frac{2(4)}{\sqrt{12}} = \frac{8}{2\sqrt{3}} = \frac{4}{\sqrt{3}}$.
None of the options match.

Since $a < b$, the major axis is along the y-axis.
The relation is $a^2 = b^2(1 - e^2)$.

Latus rectum $\frac{2b^2}{a} = 8 \Rightarrow b^2 = 4a$.
Conjugate axis $2b = \frac{1}{2}(2ae) = ae \Rightarrow 4b^2 = a^2e^2$.
$4(4a) = a^2e^2 \Rightarrow 16 = ae^2 \Rightarrow a = 16/e^2$.
Using $b^2 = a^2(e^2-1)$: $4a = a^2(e^2-1) \Rightarrow 4 = a(e^2-1)$.
$4 = \frac{16}{e^2}(e^2-1) \Rightarrow e^2 = 4e^2 - 4 \Rightarrow 3e^2 = 4 \Rightarrow e = \frac{2}{\sqrt{3}}$.

$2ae = 16 \Rightarrow ae = 8$. Given $e = \sqrt{2}$.
$a\sqrt{2} = 8 \Rightarrow a = 4\sqrt{2} \Rightarrow a^2 = 32$.
$b^2 = a^2(e^2-1) = 32(2-1) = 32$.
Equation: $\frac{x^2}{32} - \frac{y^2}{32} = 1 \Rightarrow x^2 - y^2 = 32$.

$ae = 2, e = 3/2 \Rightarrow a = 4/3 \Rightarrow a^2 = 16/9$.
$b^2 = a^2(e^2-1) = \frac{16}{9}(\frac{9}{4}-1) = \frac{20}{9}$.
Equation: $\frac{x^2}{16/9} - \frac{y^2}{20/9} = 1 \Rightarrow \frac{9x^2}{16} - \frac{9y^2}{20} = 1$.
Multiplying by 4/9: $\frac{x^2}{4} - \frac{y^2}{5} = \frac{4}{9}$.