If $x = \frac{4 λ}{1 + λ^{2}}$ and $y = \frac{2 - 2 λ^{2}}{1 + λ^{2}}$ , where $λ$ is a real parameter such that $x^{2} - x y + y^{2}$ lies in the interval $[a, b]$ , then the minimum value of $p$ for which $3 cos ϕ = p^{2} - (a + b) p + 19$ holds true is .

Open in App

Solution

Let $λ = tan θ$
So, $x = \frac{4 tan θ}{1 + {tan}^{2} θ} = 2 sin 2 θ$
Similarly $y = 2 (\frac{1 - {tan}^{2} θ}{1 + {tan}^{2} θ}) = 2 cos 2 θ$

Now, we can write
$x^{2} - x y + y^{2}$
$= 4 {sin}^{2} 2 θ - 4 sin 2 θ cos 2 θ + 4 {cos}^{2} 2 θ$
$= 4 - 2 (2 cos 2 θ sin 2 θ)$
$= 4 - 2 sin 4 θ$

We know
$- 1 \leq sin 4 θ \leq 1 \Rightarrow 2 \leq 4 - 2 sin 4 θ \leq 6$
Thus, comparing with $[a, b]$ , we get
$a = 2$ and $b = 6$
$\Rightarrow a + b = 8$
Now, we have the other equation as:
$3 cos ϕ = p^{2} - (a + b) p + 19$
$\Rightarrow 3 cos ϕ = p^{2} - 8 p + 19$
$\Rightarrow 3 cos ϕ = (p - 4)^{2} + 3$
We know that
$3 cos ϕ \leq 3$
So, $(p - 4)^{2} = 0$
$\Rightarrow p = 4$

Suggest Corrections

Similar questions

If $x = \frac{4 λ}{1 + λ^{2}}$ and $y = \frac{2 - 2 λ^{2}}{1 + λ^{2}}$ where $λ$ is a real parameter and $x^{2} - x y + y^{2}$ lies between $[a, b]$ then $(a + b)$ is

Q. If

x = \frac{4 λ}{1 + λ^{2}}

and

y = \frac{2 - 2 λ^{2}}{1 + λ^{2}}

where

λ

is a real parameter then

Z = x^{2} - x y + y^{2}

lies between

λ_{1}

and

λ_{2}

are two values of

λ

such that

f (x, y) = x^{2} + λ x y + y^{2} - 5 x - 7 y + 6

can be resolute as product of

2

linear functions &

λ_{1} > λ_{2}

. then find the value of

3 λ_{1} + 2 λ_{2}

Q. If

x = \frac{4 α}{1 + α^{2}}

and

y = \frac{2 - 2 α^{2}}{1 + α^{2}},

where

α

is a real parameter, then

x^{2} - x y + y^{2}

lines between

Q. The number of distinct real values of

λ

for which the lines

\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{λ^{2}}

and

\frac{x - 3}{1} = \frac{y - 2}{λ^{2}} = \frac{z - 1}{2}

are coplanar is:

Join BYJU'S Learning Program

If x=4λ1+λ2 and y=2−2λ21+λ2, where λ is a real parameter such that x2−xy+y2 lies in the interval [a,b], then the minimum value of p for which 3cosϕ=p2−(a+b)p+19 holds true is .

If $x = \frac{4 λ}{1 + λ^{2}}$ and $y = \frac{2 - 2 λ^{2}}{1 + λ^{2}}$ , where $λ$ is a real parameter such that $x^{2} - x y + y^{2}$ lies in the interval $[a, b]$ , then the minimum value of $p$ for which $3 cos ϕ = p^{2} - (a + b) p + 19$ holds true is .