(B) The lines are $L_1: 2x + y - 3 = 0$ and $L_2: 2x + y - 5 = 0$. These are parallel lines. For $L_1$,the intercepts are $x = 1.5$ $(A)$ and $y = 3$

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Statement-$1$ is false,Statement-$2$ is true

(B) The lines are $L_1: 2x + y - 3 = 0$ and $L_2: 2x + y - 5 = 0$. These are parallel lines. For $L_1$,the intercepts are $x = 1.5$ $(A)$ and $y = 3$ $(B)$. For $L_2$,the intercepts are $x = 2.5$ $(C)$ and $y = 5$ $(D)$. The region between the lines is given by $3 For the point $(2, 3)$,we calculate $2(2) + 3 = 4 + 3 = 7$. Since $7 > 5$,the point $(2, 3)$ does not lie between the lines. Therefore,Statement-$1$ is false. Statement-$2$ is also false because the point does not lie between the lines.

Class 11 Mathematics Straight Line Distance between two lines, Perpendicular distance of the line from a point, Position of point w.r.t. line

Difficult

Given the lines $y + 2x = 3$ and $y + 2x = 5$ cut the axes at $A, B$ and $C, D$ respectively.
Statement-$1$ : $ABDC$ forms a quadrilateral and the point $(2, 3)$ lies inside the quadrilateral.
Statement-$2$ : The point $(2, 3)$ lies between the two parallel lines.

A
Statement-$1$ is true,Statement-$2$ is true; Statement-$2$ is not the correct explanation of Statement-$1$
B
Statement-$1$ is false,Statement-$2$ is true
C
Statement-$1$ is true,Statement-$2$ is false
D
Statement-$1$ is true,Statement-$2$ is true; Statement-$2$ is the correct explanation of Statement-$1$

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Distance between two lines, Perpendicular distance of the line from a point, Position of point w.r.t. line

If $p$ and $q$ are the lengths of the perpendiculars from the origin to the straight lines $x \sec \alpha + y \operatorname{cosec} \alpha = 10$ and $x \cos \alpha - y \sin \alpha = 10 \cos 2 \alpha$,then $4 p^2 + q^2 =$

EasyAP EAMCET 2017

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If a line $L$ is perpendicular to the line $5x - y = 1$,and the area of the triangle formed by the line $L$ and the coordinate axes is $5$,then the distance of line $L$ from the line $x + 5y = 0$ is

DifficultJEE MAIN 2014

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Let $A(a, b)$,$B(3, 4)$,and $C(-6, -8)$ respectively denote the centroid,circumcentre,and orthocentre of a triangle. Then,the distance of the point $P(2a+3, 7b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is

DifficultJEE MAIN 2024

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If the points $(1, 2)$ and $(3, 4)$ lie on the same side of the straight line $3x - 5y + a = 0$,then $a$ lies in the set

DifficultTS EAMCET 2013

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The number of straight lines that can be drawn through the point $(-3, 4)$ which are at a distance of $5$ units from the point $(2, -8)$ is

MediumAP EAMCET 2023

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Given the lines $y + 2x = 3$ and $y + 2x = 5$ cut the axes at $A, B$ and $C, D$ respectively. Statement-$1$ : $ABDC$ forms a quadrilateral and the point $(2, 3)$ lies inside the quadrilateral.Statement-$2$ : The point $(2, 3)$ lies between the two parallel lines.

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If $p$ and $q$ are the lengths of the perpendiculars from the origin to the straight lines $x \sec \alpha + y \operatorname{cosec} \alpha = 10$ and $x \cos \alpha - y \sin \alpha = 10 \cos 2 \alpha$,then $4 p^2 + q^2 =$

If a line $L$ is perpendicular to the line $5x - y = 1$,and the area of the triangle formed by the line $L$ and the coordinate axes is $5$,then the distance of line $L$ from the line $x + 5y = 0$ is

Let $A(a, b)$,$B(3, 4)$,and $C(-6, -8)$ respectively denote the centroid,circumcentre,and orthocentre of a triangle. Then,the distance of the point $P(2a+3, 7b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is

If the points $(1, 2)$ and $(3, 4)$ lie on the same side of the straight line $3x - 5y + a = 0$,then $a$ lies in the set

The number of straight lines that can be drawn through the point $(-3, 4)$ which are at a distance of $5$ units from the point $(2, -8)$ is

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Given the lines $y + 2x = 3$ and $y + 2x = 5$ cut the axes at $A, B$ and $C, D$ respectively.
Statement-$1$ : $ABDC$ forms a quadrilateral and the point $(2, 3)$ lies inside the quadrilateral.
Statement-$2$ : The point $(2, 3)$ lies between the two parallel lines.