The number of integers a in the interval [1,2 | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(d)

Given,

$x+y=a \Rightarrow a \in[1,2014]$

and $\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$

$	herefore \quad \frac{x^2}{x-1}+\frac{y^2}{y-1}=4$

Vedclass · Accepted Answer

$2014$

Vedclass · Answer

(d)

Given,

$x+y=a \Rightarrow a \in[1,2014]$

and $\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$

$	herefore \quad \frac{x^2}{x-1}+\frac{y^2}{y-1}=4$

$\Rightarrow x^2 y-x^2+x y^2-y^2=4(x-1)(y-1)$

$\Rightarrow x^2 y+x y^2=4(x y-(x+y)+1)+x^2+y^2$

$\Rightarrow \quad x y(x+y)=4(x y-a+1) +(x+y)^2-2 x y$

$\Rightarrow \quad x y(a)=4 x y-4 a+4+a^2-2 x y$

$\Rightarrow \quad x y a-2 x y=a^2-4 a+4$

$\Rightarrow x y \quad(a-2)=(a-2)^2$

$\Rightarrow \quad(a-2)^2-x y(a-2)=0$

$\Rightarrow \quad(a-2)(a-2-x y)=0$

$a=2$ or $x y=a-2$

$x(a-x)=a-2 \quad[\because y=a-x]$

$x^2-a x+a-2=0$

$\Rightarrow(a 	imes b)(c+1)+b(c+1)+a(c+1)+(c+1)=30$

$\Rightarrow(c+1)(a 	imes b+b+a+1)=30$

$\Rightarrow(c+1)(b(a+1)+1(a+1))=30$

$\Rightarrow \quad(a+1))(b+1)(c+1)=30$

$1 \leq a \leq 9,0 \leq b, c \leq 9$

$\because$ Total number of solution $=18$

The number of integers $a$ in the interval $[1,2014]$ for which the system of equations $x+y=a$, $\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$ has finitely many solutions is

Similar Questions

The number of distinct real roots of the equation $x ^{7}-7 x -2=0$ is

The maximum possible number of real roots of equation ${x^5} - 6{x^2} - 4x + 5 = 0$ is

The set of values of $x$ which satisfy $5x + 2 < 3x + 8$ and $\frac{{x + 2}}{{x - 1}} < 4,$ is

Let $\alpha_1, \alpha_2, \ldots, \alpha_7$ be the roots of the equation $x^7+$ $3 x^5-13 x^3-15 x=0$ and $\left|\alpha_1\right| \geq\left|\alpha_2\right| \geq \ldots \geq\left|\alpha_7\right|$. Then $\alpha_1 \alpha_2-\alpha_3 \alpha_4+\alpha_5 \alpha_6$ is equal to $..................$.

Let $p_1(x)=x^3-2020 x^2+b_1 x+c_1$ and $p_2(x)=x^3-2021 x^2+b_2 x+c_2$ be polynomials having two common roots $\alpha$ and $\beta$. Suppose there exist polynomials $q_1(x)$ and $q_2(x)$ such that $p_1(x) q_1(x)+p_2(x) q_2(x)=x^2-3 x+2$. Then the correct identity is