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

Hint: 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 have\[h = \dfrac{{ - 3}}{2},\]\[a = 1,\]\[b = \lambda = 2\]. 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.