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Question Number 51694 by peter frank last updated on 29/Dec/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 29/Dec/18
	$point p(acosθ,bsinθ) tangent at point p is ((xcosθ)/a)+((ysinθ)/b)=1 ((ysinθ)/b)=1−((xcosθ)/a) y=(b/(sinθ))−(b/(sinθ))×((cosθ)/a)x [slope=−(b/a)cotθ] m_1 ×m_2 =−1 m_2 =((−1)/m_1 )=((−1×a)/(−bcotθ))=(a/b)tanθ so normal at point p is (y−bsinθ)=(a/b)tanθ(x−acosθ) to find point G put y=0 x−acosθ=(b/(atanθ))×−bsinθ x−acosθ=((−b^2 cosθ)/a) x=acosθ−((b^2 cosθ)/a)=((cosθ)/a)(a^2 −b^2 ) point G {((a^2 −b^2 )/a)cosθ,0} to find poit g put x=0 y−bsinθ=((atanθ)/b)(0−acosθ) y−bsinθ=((−a^2 sinθ)/b) y=bsinθ−((a^2 sinθ)/b)=((sinθ)/b)(b^2 −a^2 ) point g ={0,((sinθ)/(b ))(b^2 −a^2 )} CG=((a^2 −b^2 )/a)cosθ cg=((b^2 −a^2 )/b)sinθ a^2 CG^2 +b^2 Cg^2 =(a^2 −b^2 )^2 cos^2 θ+(b^2 −a^2 )^2 sin^2 θ =(a^2 −b^2 )^2 (cos^2 θ+sin^2 θ) =(a^2 −b^2 )^2 p(acosθ,bsinθ) N=(acosθ,0) CN=acosθ CG=((a^2 −b^2 )/a)cosθ ((CG)/(CN))=(((a^2 −b^2 )cosθ)/(a×acosθ))=((a^2 −b^2 )/a^2 ).=(c_F ^2 /a^2 )=((c_F /a))^2 =e^2 [focus(c_(F,) 0) so c_F ^2 .=a^2 −b^2 again=(c_F /a)=e]$
	$p o i n t p (a c o s θ, b s i n θ)$ $t a n g e n t a t p o i n t p i s \frac{x c o s θ}{a} + \frac{y s i n θ}{b} = 1$ $\frac{y s i n θ}{b} = 1 - \frac{x c o s θ}{a}$ $y = \frac{b}{s i n θ} - \frac{b}{s i n θ} \times \frac{c o s θ}{a} x [s l o p e = - \frac{b}{a} c o t θ]$ $m_{1} \times m_{2} = - 1$ $m_{2} = \frac{- 1}{m_{1}} = \frac{- 1 \times a}{- b c o t θ} = \frac{a}{b} t a n θ$ $s o n o r m a l a t p o i n t p i s$ $(y - b s i n θ) = \frac{a}{b} t a n θ (x - a c o s θ)$ $t o f i n d p o i n t G p u t y = 0$ $x - a c o s θ = \frac{b}{a t a n θ} \times - b s i n θ$ $x - a c o s θ = \frac{- b^{2} c o s θ}{a}$ $x = a c o s θ - \frac{b^{2} c o s θ}{a} = \frac{c o s θ}{a} (a^{2} - b^{2})$ $p o i n t G {\frac{a^{2} - b^{2}}{a} c o s θ, 0}$ $t o f i n d p o i t g p u t x = 0$ $y - b s i n θ = \frac{a t a n θ}{b} (0 - a c o s θ)$ $y - b s i n θ = \frac{- a^{2} s i n θ}{b}$ $y = b s i n θ - \frac{a^{2} s i n θ}{b} = \frac{s i n θ}{b} (b^{2} - a^{2})$ $p o i n t g = {0, \frac{s i n θ}{b} (b^{2} - a^{2})}$ $C G = \frac{a^{2} - b^{2}}{a} c o s θ$ $c g = \frac{b^{2} - a^{2}}{b} s i n θ$ $a^{2} {C G}^{2} + b^{2} {C g}^{2}$ $= {(a^{2} - b^{2})}^{2} {c o s}^{2} θ + {(b^{2} - a^{2})}^{2} {s i n}^{2} θ$ $= {(a^{2} - b^{2})}^{2} ({c o s}^{2} θ + {s i n}^{2} θ)$ $= {(a^{2} - b^{2})}^{2}$ $p (a c o s θ, b s i n θ)$ $N = (a c o s θ, 0)$ $C N = a c o s θ$ $C G = \frac{a^{2} - b^{2}}{a} c o s θ$ $\frac{C G}{C N} = \frac{(a^{2} - b^{2}) c o s θ}{a \times a c o s θ} = \frac{a^{2} - b^{2}}{a^{2}} . = \frac{c_{F}^{2}}{a^{2}} = {(\frac{c_{F}}{a})}^{2} = e^{2}$ $[f o c u s (c_{F,} 0) s o c_{F}^{2} . = a^{2} - b^{2}$ $a g a i n = \frac{c_{F}}{a} = e]$
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