Question

The line $lx + my + n = 0$ will be tangent to the hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$, if
A.${a^2}{l^2} - {b^2}{m^2} = {n^2}$
B.${a^2}{l^2} + {b^2}{m^2} = {n^2}$
C.$a{m^2} - {b^2}{n^2} = {a^2}{l^2}$
D.None of these

Answer 1

Hint: In this question, we will use the property that if $y = Mx + C$ is tangent to hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$, then ${C^2} = {a^2}{M^2} - {b^2}$ and rewrite equation of the line $lx + my + n = 0$ to compare with $y = Mx + C$ to find the value of $M$ and $C$. Then we will substitute the above values of $M$ and $C$ in the equation.

Complete step by step answer:
We are given that the line $lx + my + n = 0$ will be tangent to the hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$.

We know that if $y = Mx + C$ is tangent to hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$, then ${C^2} = {a^2}{M^2} - {b^2}$.

Rewriting the equation of the line $lx + my + n = 0$ by dividing both sides with $m$, we get

$$\Rightarrow \dfrac{{lx + my + n}}{m} = 0 \\\ \Rightarrow \dfrac{l}{m}x + \dfrac{m}{m}y + \dfrac{n}{m} = 0 \\\ \Rightarrow \dfrac{l}{m}x + y + \dfrac{n}{m} = 0 \\\$$

Subtracting the above equation by $\dfrac{l}{m}x + \dfrac{n}{m}$ on each side, we get

$$\Rightarrow y + \dfrac{l}{m}x + \dfrac{n}{m} - \dfrac{l}{m}x - \dfrac{n}{m} = - \dfrac{l}{m}x - \dfrac{n}{m} \\\ \Rightarrow y = - \dfrac{l}{m}x - \dfrac{n}{m} \\\$$

Comparing the above equation with $y = Mx + C$ to find the value of $M$ and $C$, we get

$\Rightarrow M = - \dfrac{1}{m}$
$\Rightarrow C = - \dfrac{n}{m}$

Now, substituting the above values of $M$ and $C$ in the equation ${C^2} = {a^2}{M^2} - {b^2}$, we get

$$\Rightarrow {\left( {\dfrac{n}{m}} \right)^2} = {a^2}{\left( {\dfrac{l}{m}} \right)^2} - {b^2} \\\ \Rightarrow \dfrac{{{n^2}}}{{{m^2}}} = {a^2}\dfrac{{{l^2}}}{{{m^2}}} - {b^2} \\\$$

Multiplying the above equation by ${m^2}$ on both sides, we get

$$\Rightarrow {m^2}\left( {\dfrac{{{n^2}}}{{{m^2}}}} \right) = {m^2}\left( {{a^2}\dfrac{{{l^2}}}{{{m^2}}} - {b^2}} \right) \\\ \Rightarrow {n^2} = {a^2}{l^2} - {b^2}{m^2} \\\ \Rightarrow {a^2}{l^2} - {b^2}{m^2} = {n^2} \\\$$

Hence, option A is correct.

Note: In such types of problems, students assume that the tangent meets the hyperbola at only one point, which is wrong. The other possibility of a mistake is not being able to apply the formula and properties of quadratic equations to solve. The key step to solve this problem is knowing the property that if $y = Mx + C$ is tangent to hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$, then ${C^2} = {a^2}{M^2} - {b^2}$, the solution will be very simple.