Question #224015

Find the eccentricity of the ellipse 3x2 + 4y2 = 12 and the equation of the tangent to the ellipse at the point (1, 3/2). If this tangent meets the y-axis at the point G, and S and S′ are the foci of the ellipse, find the area of triangle SS′G.


1
Expert's answer
2021-08-17T10:36:02-0400
3x2+4y2=123x^2+4y^2=12

Differentiate both sides with respect to xx


6x+8yy=06x+8yy'=0

y=3x4yy'=-\dfrac{3x}{4y}

Point (1,3/2)(1, 3/2)


y(1,32)=3(1)4(32)=12y'(1, \dfrac{3}{2})=-\dfrac{3(1)}{4(\dfrac{3}{2})}=-\dfrac{1}{2}

The equation of the tangent in point-slope form


y32=12(x1)y-\dfrac{3}{2}=-\dfrac{1}{2}(x-1)

The equation of the tangent in slope-intercept form


y=12x+2y=-\dfrac{1}{2}x+2


3x2+4y2=123x^2+4y^2=12

x24+y23=1\dfrac{x^2}{4}+\dfrac{y^2}{3}=1

a2=4,b2=3,c=a2b2=43=1a^2=4, b^2=3, c=\sqrt{a^2-b^2}=\sqrt{4-3}=1



The eccentricity of the ellipse


e=ca=12e=\dfrac{c}{a}=\dfrac{1}{2}



Foci: (±c,0)(\pm c, 0)


S(1,0),S(1,0)S(-1, 0), S'(1, 0)

x=0:y(0)=12(0)+2=2x=0: y(0)=-\dfrac{1}{2}(0)+2=2

G(0,2)G(0,2)




Triangle SGSSGS'


Area=ab(2)(1(1))=2(units2)Area= \dfrac{a}{b}(2)(1-(-1))=2 (units^2)

The area of triangle SS′G is 2 square units.



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS