Question

The straight lines joining the origin to the points of intersection of the line $2x + y = 1$ and curve $3{x^2} + 4xy - 4x + 1 = 0$ include which of the following angles?
A. $\dfrac{\pi}{ 2}$
B. $\dfrac{\pi}{3}$
C. $\dfrac{\pi}{4}$
D. $\dfrac{\pi}{6}$

Answer 1

Hint: Consider a curve and a line intersecting at two points. Now, the equation of pair of lines joining the origin and these intersection points is given by homogenizing the equation of the curve using the equation of the line.

Formula Used: The angle between a pair of straight lines, passing through the origin, represented by $a{x^2} + 2hxy + b{y^2} = 0$ is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ .

Complete step-by-step solution:
Given equation of the curve:
$3{x^2} + 4xy - 4x + 1 = 0$
Given equation of the line:
$2x + y = 1$
Now, we will obtain the equation of the pair of straight lines joining the origin and the intersection points of the given curve and line.
Hence, we will homogenize the equation of the curve using the equation of the line.
Homogenizing,
$3{x^2} + 4xy - 4x(2x + y) + 1{(2x + y)^2} = 0$
Simplifying the above equation, we get:
$3{x^2} + 4xy - 8{x^2} - 4xy + 4{x^2} + {y^2} + 4xy = 0$
On further simplification, we get:
$ - {x^2} + 4xy + {y^2} = 0$ … (1)
This is the required equation of a pair of lines formed by joining the origin and the points of intersection of the given curve and line.
General form of a pair of straight lines passing through the origin:
$a{x^2} + 2hxy + b{y^2} = 0$ … (2)
Comparing equation (1) with the general form given in equation (2),
$a = - 1$ ,
$b = 1$ and
$h = 2$
Now, we know that the tangent of the acute angle, let’s say $\theta $ , between two lines is calculated using:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
Substituting the values of the variables, we get:
$\tan \theta = \left| {\dfrac{{2\sqrt {4 + 1} }}{{( - 1) + 1}}} \right|$
On simplifying further, we get $\tan \theta = \infty $ .
Calculating the inverse, we get $\theta = \dfrac{\pi }{2}$ .
Hence, the pair of lines include an angle of \[\dfrac{\pi }{2}\] .
Thus, the correct option is A.

Note: The concept of homogenizing the equation of a curve is important in the above question. While homogenizing, you must carefully convert each term to the highest degree of the equation, by multiplying them with a term that changes its degree and is equivalent to one. Majority of the mistakes found in these questions are when the curve is being homogenized. Thus, a student must perform the multiplication correctly, to avoid any further miscalculations.