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

$\int \frac{dx}{\sqrt{x-x^2}}$ is equal to

In which of the following reactions,the concentration of product is higher than the concentration of reactant at equilibrium? $(K = \text{equilibrium constant})$

Observe the following statements:
$I$. If $dy + 2xy dx = 2e^{-x^2} dx$, then $ye^{x^2} = 2x + c$.
$II$. If $ye^{-x^2} - 2x = c$, then $dx = (2e^{-x^2} - 2xy) dy$.
Which of the following is a correct statement?

The variance of the first $50$ even natural numbers is

In the Young's double slit experiment using a monochromatic light of wavelength $\lambda$,the path difference (in terms of an integer $n$) corresponding to any point having half the peak intensity is :-

Vedclass Test Series

Exam Paper Generator

Online Exam Module