If the angle 2 theta is acute,then the acute | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The given equation is $x^2(\cos 	heta - \sin 	heta) + 2xy \cos 	heta + y^2(\cos 	heta + \sin 	heta) = 0$.Comparing this with the general form

Vedclass · Accepted Answer

$	heta$

Vedclass · Answer

(D) The given equation is $x^2(\cos 	heta - \sin 	heta) + 2xy \cos 	heta + y^2(\cos 	heta + \sin 	heta) = 0$.Comparing this with the general form $ax^2 + 2hxy + by^2 = 0$,we get:$a = \cos 	heta - \sin 	heta$,$h = \cos 	heta$,and $b = \cos 	heta + \sin 	heta$.The formula for the acute angle $\alpha$ between the pair of lines is given by $	an \alpha = \left| \frac{2 \sqrt{h^2 - ab}}{a + b} ight|$.Substituting the values:$h^2 - ab = \cos^2 	heta - (\cos 	heta - \sin 	heta)(\cos 	heta + \sin 	heta) = \cos^2 	heta - (\cos^2 	heta - \sin^2 	heta) = \sin^2 	heta$.$a + b = (\cos 	heta - \sin 	heta) + (\cos 	heta + \sin 	heta) = 2 \cos 	heta$.Thus,$	an \alpha = \left| \frac{2 \sqrt{\sin^2 	heta}}{2 \cos 	heta} ight| = \left| \frac{2 \sin 	heta}{2 \cos 	heta} ight| = |	an 	heta|$.Since $2 	heta$ is acute,$	heta$ is also acute,so $\alpha = 	heta$.

If the angle $2 \theta$ is acute,then the acute angle between the pair of straight lines $x^2(\cos \theta - \sin \theta) + 2xy \cos \theta + y^2(\cos \theta + \sin \theta) = 0$ is:

Similar Questions

In an ellipse,the distance between its foci is $6$ and its minor axis is $8$. Then its eccentricity is

If $\frac{x^3 - 6x^2 + 10x - 2}{x^2 - 5x + 6} = f(x) + \frac{A}{(x - 2)} + \frac{B}{(x - 3)}$,then $f(x) =$ ?

If $n(A) = 3$ and $n(B) = 6$ and $A \subseteq B$,then the number of elements in $A \cap B$ is equal to

If $b$ and $c$ are the lengths of the segments of any focal chord of a parabola $y^2 = 4ax$,then the length of the semi-latus rectum is:

The coils of a step-down transformer have $500$ and $5000$ turns. In the primary coil,an $AC$ of $4 \, A$ at $2200 \, V$ is sent. The value of the current and potential difference in the secondary coil will be:

Vedclass Test Series

Exam Paper Generator

Online Exam Module