Family of Lines

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JEE Main 2015

Locus of the image of the point $(2, 3)$ in the line $(2 x - 3 y + 4) + k (x - 2 y + 3) = 0, k \in R$ , is a:

2

(B)

circle of radius

3

(C)

straight line parallel to x-axis

(D)

straight line parallel to y-axis

JEE Main 2005

LEVELJEE Main

If non zero numbers $a, b, c$ are in H.P., then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point. That point is

(A)

(- 1, 2)

(B)

(- 1, - 2)

(C)

(1, - 2)

(D)

(1, - 1/2)

JEE Main 2005

LEVELJEE Main

The line parallel to the x-axis and passing through the intersection of the lines $a x + 2 b y + 3 b = 0$ and $b x - 2 a y - 3 a = 0$ , where $(a, b) \neq = (0, 0)$ is

(A)

below the x-axis at a distance of

\frac{3}{2}

from it

(B)

below the x-axis at a distance of

\frac{2}{3}

from it

(C)

above the x-axis at a distance of

\frac{3}{2}

from it

(D)

above the x-axis at a distance of

\frac{2}{3}

from it

JEE Advanced 1985

LEVELJEE Main

Three lines $p x + q y + r = 0, q x + r y + p = 0$ and $r x + p y + q = 0$ are concurrent if

* Multiple Correct Options

(A)

p + q + r = 0

(B)

p^{2} + q^{2} + r^{2} = q r + r p + pq

(C)

p^{3} + q^{3} + r^{3} = 3 pq r

Let the algebraic sum of the perpendicular distances from the points $(2, 0), (0, 2)$ and $(1, 1)$ to a variable straight line be zero; then the line passes through a fixed point whose coordinates are .........

JEE Advanced 1991

LEVELJEE Advanced

Show that all chords of the curve $3 x^{2} - y^{2} - 2 x + 4 y = 0$ , which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

JEE Advanced 1988

LEVELJEE Main

Lines $L_{1} \equiv a x + b y + c = 0$ and $L_{2} \equiv l x + m y + n = 0$ intersect at the point $P$ and make an angle $θ$ with each other. Find the equation of a line $L$ different from $L_{2}$ which passes through $P$ and makes the same angle $θ$ with $L_{1}$ .

JEE Advanced 1984

LEVELJEE Main

If $a, b$ and $c$ are in A.P., then the straight line $a x + b y + c = 0$ will always pass through a fixed point whose coordinates are .........

JEE Advanced 1982

LEVELJEE Main

The set of lines $a x + b y + c = 0$ , where $3 a + 2 b + 4 c = 0$ is concurrent at the point .........

JEE Main 2019 (9 January)

LEVELJEE Main

Consider the set of all lines $p x + q y + r = 0$ such that $3 p + 2 q + 4 r = 0$ . Which one of the following statements is true ?

(A)

The lines are all parallel.

(B)

Each line passes through the origin.

(C)

The lines are not concurrent.

(D)

The lines are concurrent at the point

(3, 1/2)

JEE Main 2025 (April)

LEVELJEE Advanced

Consider the lines $x (3 λ + 1) + y (7 λ + 2) = 17 λ + 5, λ$ being a parameter, all passing through a point $P$ . One of these lines (say $L$ ) is farthest from the origin. If the distance of $L$ from the point $(3, 6)$ is $d$ , then the value of $d^{2}$ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Locus of the image of the point (2,3) in the line (2x−3y+4)+k(x−2y+3)=0,k∈R, is a:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If non zero numbers a,b,c are in H.P., then the straight line ax​+by​+c1​=0 always passes through a fixed point. That point is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The line parallel to the x-axis and passing through the intersection of the lines ax+2by+3b=0 and bx−2ay−3a=0, where (a,b)=(0,0) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Three lines px+qy+r=0,qx+ry+p=0 and rx+py+q=0 are concurrent if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Show that all chords of the curve 3x2−y2−2x+4y=0, which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b and c are in A.P., then the straight line ax+by+c=0 will always pass through a fixed point whose coordinates are .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of lines ax+by+c=0, where 3a+2b+4c=0 is concurrent at the point .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the set of all lines px+qy+r=0 such that 3p+2q+4r=0. Which one of the following statements is true ?

Locus of the image of the point $(2, 3)$ in the line $(2 x - 3 y + 4) + k (x - 2 y + 3) = 0, k \in R$ , is a:

If non zero numbers $a, b, c$ are in H.P., then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point. That point is

The line parallel to the x-axis and passing through the intersection of the lines $a x + 2 b y + 3 b = 0$ and $b x - 2 a y - 3 a = 0$ , where $(a, b) \neq = (0, 0)$ is

Three lines $p x + q y + r = 0, q x + r y + p = 0$ and $r x + p y + q = 0$ are concurrent if

Show that all chords of the curve $3 x^{2} - y^{2} - 2 x + 4 y = 0$ , which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

If $a, b$ and $c$ are in A.P., then the straight line $a x + b y + c = 0$ will always pass through a fixed point whose coordinates are .........

The set of lines $a x + b y + c = 0$ , where $3 a + 2 b + 4 c = 0$ is concurrent at the point .........

Consider the set of all lines $p x + q y + r = 0$ such that $3 p + 2 q + 4 r = 0$ . Which one of the following statements is true ?