Question

What is the acute angle formed between the lines joining the origin to the points of intersection of the curve ${x^2} + {y^2} - 2x - 1 = 0$ and the line $x + y = 1$ ?
A. \[{\tan ^{ - 1}}\left( { - \dfrac{1}{2}} \right)\]
B. \[{\tan ^{ - 1}}2\]
C. \[{\tan ^{ - 1}}\left( {\dfrac{1}{2}} \right)\]
D. ${60^ \circ }$

Answer 1

Hint: Consider a curve and a line intersecting at two points. Now, the equation of a 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. Then using the angle formula we will get the required solution.

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:
${x^2} + {y^2} - 2x - 1 = 0$
Given equation of the line:
$x + y = 1$
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,
${x^2} + {y^2} - 2x(x + y) - 1{(x + y)^2} = 0$
Simplifying the above equation, we get:
${x^2} + {y^2} - 2{x^2} - 2xy - {x^2} - {y^2} - 2xy = 0$
On further simplification, we get:
$ - 2{x^2} - 4xy = 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),
$a = - 2$ ,
$b = 0$ 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:
$\tan \theta = \left| {\dfrac{{2\sqrt {4 - 0} }}{{ - 2}}} \right|$
On simplifying further, we get $\tan \theta = 2$ .
Calculating the inverse, we get $\theta = {\tan ^{ - 1}}(2)$ .
Hence, the angle between the pair of lines is \[{\tan ^{ - 1}}2\] .
Thus, the correct option is B.

Note: Homogenizing the equation of the curve to get the required equation is an important step in the above question. Avoid making any mistakes while homogenizing. Further, substitute the correct values in the formula to calculate the tangent of the angle between the pair of lines.