Question

If x ^ 2 = 8ay is the transformed equation of x ^ 2 - 4y + 6x + 15 = 0 when the origin is shifted to the point (alpha, beta) by translation of axes, then 2alpha + 8beta ^ 2 = (a) 8 (b) 18 (c) 12 (d) 16

Answer 1

https://www.sarthaks.com/?qa=blob&qa_blobid=6015188532224823635

When origin shifted to ( h.k) Then x→ x+h 

y → y +k 

Substituting new x and y values in the equation

(x+h)² - 4 (y +k ) + 6 (x+h ) + 15 = 0 

x² + h² +2hx – 4y -4k +6x + 6h + 15 = 0 

x² + 2hx – 4y +6x + h2 + 6h -4k + 15 = 0 

x² + (2h + 6) x – 4y + h² + 6h -4k + 15 = 0

there is no x term in final Equation, then 2h+6= 0 then 

h = -3

x² – 4y + (-3)² + 6(-3) -4k + 15 = 0 

x² – 4y + 9 -18 -4k + 15 = 0 

x² – 4y + 6 -4k = 0 

there is no constant term in final Equation, then 6 -4k= 0 

then k = 3/2 

Origin is shifted to ( h.k ) = (-3, 3/2) 

2h + 8 k² = 2(-3) + 8 (9/4) = -6 +18 =12