The value of `lambda` for which the curve `(7x + 5)^2 + (7y + 3)^2 = lambda^2 (4x + 3y-24)^2` represents a parabola is

Options

A. `+-(6)/(5)`
B. `+-(7)/(5)`
C. `+-(1)/(5)`
D. `+-(2)/(5)`

Answer

B

Explanation

Given curve is `(7x+5)^2+(7y+3)^2`
`=lambda^2(4x+3y-24y)^2`
`rArr 49x^2+25+70x+49y^2+9+42y`
`lambda^2(16x^2+9y^2+576+24xy-144y-192x)`
`rArr (49-16lambda^2)x^2+(39-9lambda^2)y^2+(70+192lambda^2)x+(42+144lambda^2)y-24lambda^2xy+(25-576lambda^2)=0`
On comparing with
`ax^2+2hxy+by^2+2gh+2fy+c=0` we get
`a=49-16lambda^2,b=(49-9lambda^2)`
`h=-12lambda^2`
Condition for parabola is `ab=h^2`
`therefore (49-16lambda^2)(49-9lambda^2)=(-12lambda^2)^2`
`rArr 2401-441lambda^2-784lambda^2+144lambda^4-144lambda^4=0`
`rArr 2401-1225lambda^2=0`
`rArr lambda^2=(2401)/(1225)rArr lambda^2=((49)^2)/((35)^2)`
`rArr lambda=+-(49)/(35)`
`therefore lambda=+-(7)/(5)`