If the lines 2x + 3y + 12 = 0 and x - y + k = | vedclass

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

Question

(C) Given,the conjugate lines are $2x + 3y + 12 = 0$ $(i)$ and $x - y + k = 0$ $(ii)$.Two lines are conjugate with respect to a parabola if the pole o

Vedclass · Accepted Answer

$-12$

Vedclass · Answer

(C) Given,the conjugate lines are $2x + 3y + 12 = 0$ $(i)$ and $x - y + k = 0$ $(ii)$.Two lines are conjugate with respect to a parabola if the pole of one line lies on the other line.Let the pole of the line $2x + 3y + 12 = 0$ be $(x_1, y_1)$ with respect to the parabola $y^2 = 8x$.The equation of the polar of the point $(x_1, y_1)$ with respect to $y^2 = 8x$ is $yy_1 = 4(x + x_1)$,which simplifies to $4x - yy_1 + 4x_1 = 0$.Comparing this with the given line $2x + 3y + 12 = 0$,we have:$\frac{4}{2} = \frac{-y_1}{3} = \frac{4x_1}{12}$$2 = \frac{-y_1}{3} \Rightarrow y_1 = -6$$2 = \frac{x_1}{3} \Rightarrow x_1 = 6$Thus,the pole is $(6, -6)$.Since the lines are conjugate,the pole $(6, -6)$ must lie on the second line $x - y + k = 0$.Substituting the coordinates,we get $6 - (-6) + k = 0$.$12 + k = 0 \Rightarrow k = -12$.

If the lines $2x + 3y + 12 = 0$ and $x - y + k = 0$ are conjugate with respect to the parabola $y^2 = 8x$,then $k$ is equal to

Similar Questions

$A$ $4 \, kg$ mass and a $1 \, kg$ mass are moving with equal kinetic energies. The ratio of the magnitudes of their linear momenta is

The $INCORRECT$ statement about the dissolution of an alkali metal in liquid ammonia is

Which genotype represents a true dihybrid condition?

$A$ tower subtends angles $\alpha, 2 \alpha$ and $3 \alpha$ respectively at points $A, B$ and $C$,all lying on a horizontal line through the foot of the tower. Then $\frac{A B}{B C}$ is equal to:

$A$ river is flowing from east to west at a speed of $5\ m/min$. $A$ man on the south bank of the river,capable of swimming $10\ m/min$ in still water,wants to swim across the river in the shortest time. He should swim:

Vedclass Test Series

Exam Paper Generator

Online Exam Module