The lines $p\left(p^{2}+1\right) x-y+q=0$ and $\left(p^{2}+1\right)^{2} x+\left(p^{2}+1\right) y+2 q=0$ are perpendicular to a common line for

(C) exactly two values of $p$

(D) more than two values of $p$

JEE Main · 2009 · 12 Apr 2009 · Offline · Straight Lines

The lines

p (p^{2} + 1) x - y + q = 0

and

(p^{2} + 1)^{2} x + (p^{2} + 1) y + 2 q = 0

are perpendicular to a common line for

(A)no value of p
(B)exactly one value of $p$ ✓ Correct
(C)exactly two values of $p$
(D)more than two values of $p$

Answer: (B)

Solution

Two lines that are both perpendicular to one common line must be parallel to each other, so their slopes have to be equal. The slopes here are

p (p^{2} + 1)

and

- \frac{( p ^{2} + 1 ) ^{2}}{p ^{2} + 1} = - (p^{2} + 1)

, and setting them equal gives

p (p^{2} + 1) = - (p^{2} + 1)

. Dividing by the nonzero factor

p^{2} + 1

leaves

p = - 1

, so there is exactly one value of

p

JEE Main · 2009 · 12 Apr 2009 · Offline · Single correct option

Practise more: all JEE Main PYQs on Straight Lines →

More JEE Main questions on Straight Lines

Let a point A lie between the parallel lines $L_{1}$ and $L_{2}$ such that its distances from $L_{1}$ and $L_{2}$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C l …
JEE Main 2026View solution →
Among the statements
(S1): If $A (5, - 1)$ and $B (- 2, 3)$ are two vertices of a triangle, whose orthocentre is $(0, 0)$ , then its third vertex is $(- 4, - 7)$ and
(S2): If positive numbers $2 a, b, c$ are three consecutive terms of an A.P., then the lines $a x + b y + c = 0$ are …
JEE Main 2026View solution →
A rectangle is formed by the lines $x = 0$ , $y = 0$ , $x = 3$ and $y = 4$ . Let the line L be perpendicular to $3 x + y + 6 = 0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $(\frac{1}{2}, - 5)$ …
JEE Main 2026View solution →
Let $A (1, 2)$ and $C (- 3, - 6)$ be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line $7 x - y = 14$ . If $B (α, β)$ and $D (γ, δ)$ are the other two vertices, then …
JEE Main 2026View solution →
Let $A (1, 0), B (2, - 1)$ and $C (\frac{7}{3}, \frac{4}{3})$ be three points. If the equation of the bisector of the angle ABC is $α x + β y = 5$ , then the value of $α^{2} + β^{2}$ is
JEE Main 2026View solution →
Let the angles made with the positive x-axis by two straight lines drawn from the point $P (2, 3)$ and meeting the line $x + y = 6$ at a distance $\frac{2}{3}$ from the point P be $θ_{1}$ and $θ_{2}$ . Then the value of …
JEE Main 2026View solution →