Question

The equation ${x^2} - 3xy + \lambda {y^2} + 3x - 5y + 2 = 0$ when $\lambda$ is a real number, represents a pair of straight lines. If $\theta$ is the angle between the lines, then ${\csc ^2}\theta$ is equal to
a). $3$
b). $9$
c). $10$
d). $100$

Answer 1

Here we are asked to find the value of ${\csc ^2}\theta$ where $\theta$ is the angle between the given pair of straight lines. First, we will find the components of the equation of a pair of straight lines by comparing it with the general equation. Then we will find the value of the unknown term $\lambda$ . After that, we will find the angle between the lines which can be modified to find the required value.
Formula: Some formulas that we will be using in this problem:
If $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ be the pair of straight lines then $abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$
The angle between the pair of straight lines,$\tan \theta = \dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}$ where $\theta$ is the angle between them.
$\dfrac{1}{{\tan \theta }} = \cot \theta$
$1 + {\cot ^2}\theta = {\csc ^2}\theta$

Complete step-by-step solution:
It is given that ${x^2} - 3xy + \lambda {y^2} + 3x - 5y + 2 = 0$ is a pair of straight lines where $\lambda$is a real number. We aim to find the value of ${\csc ^2}\theta$ where $\theta$ is the angle between the given pair of straight lines.
We know that the generalized form of a pair of straight lines is $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$.
Comparing this and the given equation we get
a = 1,$$$$h = \dfrac{{ - 3}}{2},$$$$b = \lambda ,$$$$g = \dfrac{3}{2},$$$$f = \dfrac{{ - 5}}{2},$$$$c = 2
Thus, we have found all the components of the equation of a pair of straight lines.
We know that $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ be the pair of straight lines then $abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$
Let’s substitute the components that we found in the expression$abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$.
$abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$ $\Rightarrow 2\lambda + \dfrac{{45}}{4} - \dfrac{{25}}{4} - \dfrac{{9\lambda }}{4} - \dfrac{9}{2} = 0$
On simplifying the above equation, we get
$\Rightarrow \dfrac{{8\lambda + 45 - 25 - 9\lambda - 18}}{4} = 0$
Let us simplify it further.
$\Rightarrow 8\lambda + 45 - 25 - 9\lambda - 18 = 0$
$\Rightarrow 2 - \lambda = 0$
$\Rightarrow \lambda = 2$
Thus, we have found the value of the real number$\lambda = 2$. Now let us find the angle between the pair of straight lines.
We know that if the angle between the lines is$\theta$, then$\tan \theta = \dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}$
Here we haveh = \dfrac{{ - 3}}{2},$$$$a = 1,$$$$b = \lambda = 2. Substituting these values, we get
$\tan \theta = \dfrac{{2\sqrt {{{\left( {\dfrac{{ - 3}}{2}} \right)}^2} - (1)(2)} }}{{1 + 2}}$
On simplifying this we get
$\tan \theta = \dfrac{{2\sqrt {\dfrac{9}{4} - 2} }}{3}$
On simplifying it further we get
$\theta$
$\tan \theta = \dfrac{{2\left( {\dfrac{1}{2}} \right)}}{3}$
$\tan \theta = \dfrac{1}{3}$
Let us reciprocal the above equation.
$\dfrac{1}{{\tan \theta }} = 3$
Using the formula $\dfrac{1}{{\tan \theta }} = \cot \theta$we get
$\cot \theta = 3$
Squaring the above equation, we get
${\cot ^2}\theta = 9$
Now let us substitute the above value in $1 + {\cot ^2}\theta = {\csc ^2}\theta$
$\Rightarrow {\csc ^2}\theta = 1 + 9 = 10$
Thus, we got the value of${\csc ^2}\theta = 10$. Now let us see the options to find the correct answer.
Option (a) $3$ is an incorrect answer since we got that ${\csc ^2}\theta = 10$ in our calculation.
Option (b) $9$ is an incorrect answer since we got that ${\csc ^2}\theta = 10$ in our calculation.
Option (c) $10$ is the correct answer as we got the same value in our calculation above.
Option (d) $100$ is an incorrect answer since we got that ${\csc ^2}\theta = 10$ in our calculation.
Hence, option (c) $10$ is the correct answer.

Note: In this problem, it was necessary to find the value of each component in the equation of a pair of straight lines since that will be used to find the angle between them using the formula. Here they asked to find the value of ${\csc ^2}\theta$ since $\theta$ is the angle between the pair of straight lines we first found the $\tan \theta$ value using the standard formula and modified it using the trigonometric identity to find the required value.