The condition for the lines lx + my + n = 0 a | vedclass

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

Question

(A) Two lines $L_1: lx + my + n = 0$ and $L_2: l_1x + m_1y + n_1 = 0$ are conjugate with respect to the circle $x^2 + y^2 = r^2$ if the pole of the fi

Vedclass · Accepted Answer

$r^2(ll_1 + mm_1) = nn_1$

Vedclass · Answer

(A) Two lines $L_1: lx + my + n = 0$ and $L_2: l_1x + m_1y + n_1 = 0$ are conjugate with respect to the circle $x^2 + y^2 = r^2$ if the pole of the first line lies on the second line.The pole of the line $lx + my + n = 0$ with respect to the circle $x^2 + y^2 = r^2$ is given by the point $(x_0, y_0)$,where $x_0 = -\frac{lr^2}{n}$ and $y_0 = -\frac{mr^2}{n}$.Since this point lies on the line $l_1x + m_1y + n_1 = 0$,we have:$l_1(-\frac{lr^2}{n}) + m_1(-\frac{mr^2}{n}) + n_1 = 0$Multiplying by $-n$,we get:$l_1lr^2 + m_1mr^2 = nn_1$$r^2(ll_1 + mm_1) = nn_1$

The condition for the lines $lx + my + n = 0$ and $l_1x + m_1y + n_1 = 0$ to be conjugate with respect to the circle $x^2 + y^2 = r^2$ is

Similar Questions

The atomic numbers of Vanadium $(V)$,Chromium $(Cr)$,Manganese $(Mn)$,and Iron $(Fe)$ are $23, 24, 25$,and $26$ respectively. Which of the following has the highest second ionization enthalpy?

$A$ tension of $20 ~N$ is applied to a copper wire of cross-sectional area $0.01 ~cm^2$. The Young's modulus of copper is $1.1 \times 10^{11} ~N/m^2$ and the Poisson's ratio is $0.32$. The decrease in the cross-sectional area of the wire is:

$A$ gas mixture consists of $2$ moles of oxygen and $4$ moles of argon at temperature $T$. Neglecting all vibrational modes,the total internal energy of the system is: (in $RT$)

$A$ Zener diode,having a breakdown voltage equal to $15 \, V$,is used in a voltage regulator circuit as shown in the figure. The current through the diode is.....$mA$.

Identify the logic operation carried out by the following circuit.

Vedclass Test Series

Exam Paper Generator

Online Exam Module