If P(alpha, beta) is a point on the line 3x + | vedclass

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

Question

(A) Let the line be $L(x, y) = 3x - 4y - 8 = 0$. For point $Q(1, 1)$,$L(1, 1) = 3(1) - 4(1) - 8 = 3 - 4 - 8 = -9$. Since $L(1, 1) < 0$,for $P(\alpha,

Vedclass · Accepted Answer

$\alpha > \frac{8}{15}, \beta < -\frac{8}{5}$

Vedclass · Answer

(A) Let the line be $L(x, y) = 3x - 4y - 8 = 0$. For point $Q(1, 1)$,$L(1, 1) = 3(1) - 4(1) - 8 = 3 - 4 - 8 = -9$. Since $L(1, 1)  0$. Substituting $P(\alpha, \beta)$ into the line equation: $3\alpha - 4\beta - 8 > 0$. Since $P$ lies on $3x + y = 0$,we have $\beta = -3\alpha$. Substituting $\beta = -3\alpha$ into the inequality: $3\alpha - 4(-3\alpha) - 8 > 0$. $3\alpha + 12\alpha - 8 > 0$ $\Rightarrow 15\alpha > 8$ $\Rightarrow \alpha > \frac{8}{15}$. Now,using $\alpha = -\frac{\beta}{3}$ in the inequality $3\alpha - 4\beta - 8 > 0$: $3(-\frac{\beta}{3}) - 4\beta - 8 > 0$ $\Rightarrow -\beta - 4\beta - 8 > 0$ $\Rightarrow -5\beta > 8$ $\Rightarrow \beta Thus,$\alpha > \frac{8}{15}$ and $\beta < -\frac{8}{5}$.

If $P(\alpha, \beta)$ is a point on the line $3x + y = 0$ such that the point $P$ and the point $Q(1, 1)$ lie on opposite sides of the line $3x - 4y - 8 = 0$,then:

Similar Questions

Find the equation of the line which is equidistant from the parallel lines $9x + 6y - 7 = 0$ and $3x + 2y + 6 = 0$.

The points on the $x$-axis whose perpendicular distance from the line $\frac{x}{a} + \frac{y}{b} = 1$ is $a$ are

The line $L_1$ given by $\frac{x}{p} + \frac{y}{2} = 1$ passes through the point $(5, 0)$. The line $L_2$ given by $\frac{x}{10} + \frac{y}{q} = 1$ is parallel to $L_1$. Find the distance between the lines $L_1$ and $L_2$.

An equilateral triangle is constructed between the lines $\sqrt{3} x+y-6=0$ and $\sqrt{3} x+y+9=0$ with the base on one line and the vertex on the other. The area (in sq. units) of the triangle so formed is

If the point $(a, a)$ lies between the lines $|x + y| = 2$,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module