If the pair of lines

ax2 + 2hxy+ by2 + 2gx + 2fy+ c = 0 intersect on the y - axis then2+ch2 b) bg2 ≠ ch2 c) abc = 2fgh d) none of these

Please give me a detailed solution with reason rates

Ans a

a) 2fgh = bg^

a) 2fgh = bg

Nice problem dude ,

ax ² + by² + 2hxy +  2gx + 2fy + c = 0 ...........................................1

Condition for a pair of straight lines is

abc + 2fgh - af ²  - bg ² - ch ² = 0 .................................................2

Now , let the point of intersection on the y axis be ( 0,t )

Now , S / x = 2ax + 2hy + 2g = 0

S / y = 0 = 2hx + 2by + 2f = 0

So ht + g = 0 , bt + f = 0 [ as x = 0 , it lies on the Y axis ]

So , now we have hf = bg ................................................................3

The equation of Y axis is x = 0

Now solving this and the equation of pair of lines, we get

Now the given equation is , ax ² + by² + 2hxy +  2gx + 2fy + c = 0

So putting x = 0 , we have by ² + 2fy + c = 0

So , they intersect at one point , hence they should be having only 1 root

So applying that , ( D = 0 ) , we have  f ² = bc ...............................4

So we have from eqn 3 , fh = bg

or fgh = bg ² ........................................................................................5

Now look at 2 ,

we have abc + 2fgh - af ²  - bg ² - ch ² = 0 

So using 4 and 5

abc + 2fgh - fgh - abc - ch² = 0

So we have ch ² = fgh

So the following things can be deduced

fgh = ch² = bg²

and 2fgh = bg ² + ch² 

Too difficult to understand your options!!

Hence solved

[ P.S : I didnt find your question in my AIEEE 10 MOCK TESTs BOOK .~ Arihant Publishers I have question papers  and solutions upto 2006 as I bought it in Class 11.Moreover its not copy pasted , impossible to find such a detailed solution in those books ! ]