The set of values of b for which the origin a | vedclass

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

Question

(B) Let the line be $L(x, y) = a^2x + aby + 1 = 0$.For the origin $(0, 0)$,$L(0, 0) = 0 + 0 + 1 = 1$,which is positive.For the points to lie on the sa

Vedclass · Accepted Answer

$b \in (0, 2)$

Vedclass · Answer

(B) Let the line be $L(x, y) = a^2x + aby + 1 = 0$.For the origin $(0, 0)$,$L(0, 0) = 0 + 0 + 1 = 1$,which is positive.For the points to lie on the same side,$L(1, 1)$ must also be positive for all $a \in R$.$L(1, 1) = a^2(1) + ab(1) + 1 = a^2 + ab + 1 > 0$.This is a quadratic expression in $a$ with a positive leading coefficient $(1 > 0)$.For this to be positive for all $a \in R$,the discriminant $D$ must be less than $0$.$D = (b)^2 - 4(1)(1) $b^2 - 4 $(b - 2)(b + 2) This implies $-2 Given that $b > 0$,the intersection of these conditions is $b \in (0, 2)$.

The set of values of $b$ for which the origin and the point $(1, 1)$ lie on the same side of the straight line $a^2x + aby + 1 = 0$ for all $a \in R$ and $b > 0$ is:

Similar Questions

If $P_1$ and $P_2$ are the lengths of the perpendiculars from the origin to the lines $x \sec \theta + y \csc \theta = a$ and $x \cos \theta - y \sin \theta = a \cos 2\theta$ respectively,then what is the value of $4P_1^2 + P_2^2$?

From the following options,the line nearest to the origin is ....

If the points $(1, 2)$ and $(3, 4)$ lie on opposite sides of the line $3x - 5y + a = 0$,then:

If $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line $x+y+3=0$ such that each of them is at a distance of $\sqrt{5}$ units from the line $x+2y+2=0$,then the value of $|x_1-x_2|$ is:

The coordinates of a point on the line $x+y+1=0,$ which is at a distance of $\frac{1}{5}$ unit from the line $3x+4y+2=0,$ are

Vedclass Test Series

Exam Paper Generator

Online Exam Module