The origin belongs to a region between the li | vedclass

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

Question

(B) Let $L_1(x, y) = x+2y-5$ and $L_2(x, y) = 3x-4y+5$. The origin $(0,0)$ gives $L_1(0,0) = -5$ and $L_2(0,0) = 5$. Since they have opposite signs,th

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(B) Let $L_1(x, y) = x+2y-5$ and $L_2(x, y) = 3x-4y+5$. The origin $(0,0)$ gives $L_1(0,0) = -5$ and $L_2(0,0) = 5$. Since they have opposite signs,the origin lies between the lines.For a point $P = ((\alpha-1)^2, \alpha)$ to lie in the same region as the origin,$L_1(P)$ must have the same sign as $L_1(0,0)$,i.e.,$L_1(P)  0$.Condition $1$: $(\alpha-1)^2 + 2\alpha - 5 Condition $2$: $3(\alpha-1)^2 - 4\alpha + 5 > 0$ $\Rightarrow 3(\alpha^2 - 2\alpha + 1) - 4\alpha + 5 > 0$ $\Rightarrow 3\alpha^2 - 10\alpha + 8 > 0$ $\Rightarrow (3\alpha-4)(\alpha-2) > 0$ $\Rightarrow \alpha  2$.Combining the conditions: $\alpha \in (-2, 2) \cap (\alpha  2) \Rightarrow \alpha \in (-2, \frac{4}{3})$.Since $\alpha \in \mathbb{Z}$,the possible values are $\alpha \in \{-1, 0, 1\}$.Thus,there are $3$ such points.

The origin belongs to a region between the lines $x+2y-5=0$ and $3x-4y+5=0$. The number of points in that region of the form $((\alpha-1)^2, \alpha)$,where $\alpha \in \mathbb{Z}$,is

Similar Questions

Find the solution set of the inequality: $4x + 3 \geq 2x + 17$ and $3x - 5 < -2$.

$A$ solution of $8 \%$ boric acid is to be diluted by adding a $2 \%$ boric acid solution to it. The resulting mixture is to be more than $4 \%$ but less than $6 \%$ boric acid. If we have $640$ litres of the $8 \%$ solution,how many litres of the $2 \%$ solution will have to be added?

$\left\{x \in R: \frac{14 x}{x+1}-\frac{9 x-30}{x-4} < 0\right\}$ is equal to

The length of a rectangle is five times the breadth. If the minimum perimeter of the rectangle is $180 \ cm$,then:

$A$ solution of $9 \%$ acid is to be diluted by adding $3 \%$ acid solution to it. The resulting mixture is to be more than $5 \%$ but less than $7 \%$ acid. If there is $460 \text{ L}$ of the $9 \%$ solution,how many litres of $3 \%$ solution will have to be added?

Vedclass Test Series

Exam Paper Generator

Online Exam Module