Question

The reflection of the curve xy=1 in the line y = 2x is the curve 12x square +rxy+sy square +t=0 then the
value of 'r' is
1-7
2) 25
3) - 175
4) 90​

Answer 1

Answer: The correct answer to the above question is option (1) -7 .

Step-by-step explanation:

Let image of point A(α,β) about y=2x is B(a,b)

mid point of AB i.e. M = (α+a)/2 , (β+b)/2) lie on y=2x

Putting these values on the equation of line we get,

(β+b) / 2 = 2 ((α+a)/2)

β+b = 2α + 2a    ..................... equation 1

Slope of AB = $\frac{-1}{2}$

which means

(β-b)  /  (α-a) = $\frac{-1}{2}$

∴ (β-b) = a/2 - α/2   ..........................equation 2

Subtracting equation 1 and 2, we get

2b = $\frac{5}{2}$ α + $\frac{3}{2}$ α

α = $\frac{4b-3a}{5}$ ...........................equation 3

β = $\frac{8b-6a}{5}$ + 2a-b

β= $\frac{3b+4a}{5}$  ..............................  equation 4

(α,β) lies on xy=1

Substituting values obtained in equation 3 and 4 in the equation xy=1

( $\frac{4b-3a}{5}$ )( $\frac{3b+4a}{5}$ ) = 1

12$b^2$ + 7ab −12$a^2$ = 25

12$x^{2}$ - 7xy = 12$y^{2}$ + 25 = 0

Hence, the Value of r = -7.

Answer 2

The  value of 'r' is 1) -7

Step-by-step explanation:

The image of the point A(α, β) about y = 2x be B(a, b)

The mid point of AB lying on y = 2x  is given as:

M = (α + a)/2, (β + b)/2

On substituting the values on the equation of line, we get,

(β + b)/2 = 2 ((α + a)/2)

∴ β + b = 2α + 2a → (equation 1)

Slope of AB = -1/2 = (β - b)/(α - a)

∴ (β - b) = a/2 - α/2 → (equation 2)

On subtracting equation (1) and (2), we get

2b = 5/2α + 3/2α

∴ α = (4b - 3a)/5 → (equation 3)

β = (8b - 6a)/5 + 2a - b

∴ β = (3b + 4a)/5 → (equation 4)

From question, (α, β) lies on xy = 1. So, substituting values in equation (3) and (4) in the equation xy = 1

(4b-3a)/5 × (3b + 4a)/5 = 1

12b² + 7ab - 12a² = 25

∴ 12x² - 7xy = 12y² + 25 = 0