Question

For all values of $\theta$ the locus of the point of intersection of the lines $x \cos \theta+y \sin \theta=a$ and $x \sin \theta-y \cos \theta=b$ is
(A) An ellipse
(B) A circle
(c) A parabola
(D) A hyperbola

Answer 1

Hint:We have to find the point of intersection of the given lines. For that, we are squaring the equations and comparing both the resultant equations to find the resultant equation. After that, we have to check the equation whether it is an ellipse or a circle or a parabola, hyperbola.

Complete step by step Solution:
Given that
$x\cos \theta +y\sin \theta =a\text{ }..........\text{(1)}$
$x\sin \theta -y\cos \theta =b\text{ }................\text{(2)}$
When we square and add (1) and (2), we get
$x^{2}+y^{2}=a^{2}+b^{2}$
This is the equation of a circle
Therefore, the correct option is B.

Additional Information:A locus is a curve or other shape that is created by a point, line, or moving surface, or by all the points that meet a specific equation of the relation between the coordinates. All shapes, including circles, ellipses, parabolas, and hyperbolas, are defined by the locus as collections of points. We can write the equation for a circle if we know the location of the circle's center and how long its radius is. All of the points on the circle's circumference are represented by the circle equation. The group of points whose separation from a fixed point has a constant value is represented by a circle. The radius of the circle abbreviated $r$, is a constant that describes this fixed point, which is known as the circle's center.

Note:Students should keep in that the general equation for a circle is $\left(\mathrm{x}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}-\mathrm{y}_{1}\right)^{2}=\mathrm{r}^{2}$ whose centre is at $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$