The coordinates of the middle point of the chord intercepted on the line `2x-y+3=0` by the ellipse `(x^(2))/(10)+(y^(2))/(6)=1` are

Options

A. `((-30)/(23),(9)/(23))`
B. `(-1,1)`
C. `(-2,-1)`
D. none of these

Answer

A

Explanation

the equation of the ellipse
`(x^(2))/(10)+(y^(2))/(6)=1 or 3x^(2)+5y^(2)=30`
Let (h,k) be the mid-point of the chord intercepted on the line ` 2x-y+3=0` by the ellipse (i) then equation of the chord is
`T=S'`
`i.e., 3hx+5ky -30 =3h^(2) +5k^(2)-30`
` or 3hx _ 5ky = 3h^(2)+5k^(2)`
the equation of the line is
`2x-y+3=0`
Clearly (ii) and (iii) represent the same line .
`therefore (3h)/(2)=-5k=(3h^(2)+5k^(2))/(-3)=lamda ("say")`
`implies h = (2lmada )/( 3),k =(-lamda)/(5) and 3h^(2)+5k^(2)=--3lamda `
`implies 3((4lamda^(2))/(9))+5((lamda^(2))/(25))=-3lamda `
`implies ( 4 lamda^(2))/(3)+(lamda^(2))/(5)=-3 lamdaimplies 23lamda^(2)=-45lamda implies lamda =0, lamda=(-45)/(23)`
But `lamda=0` is not possible So, `lamda=(-45)/(23)`
`Now, h =(2lamda )/(3),k=(-lamda)/(5)implies h=-(30)/(23).k=(9)/(23)`
hence , the required coordinates of the mid -point are
`(-30//23,9//23)`