Circle And System Of Circles Question 84
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
Options:
A) An ellipse is a closed curve traced by a point moving in a plane such that the sum of the distances from two fixed points is constant.
B) A circle
C) A parabola is a U-shaped curve where any point is at an equal distance from the focus and directrix. It is defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). The standard form of a parabola that opens upward is $ y = ax^2 + bx + c $, where $ a $ determines the width and direction of the parabola.
D) A hyperbola is a conic section defined as the set of all points in a plane where the absolute difference of the distances from two fixed points (foci) is constant.
Show Answer
Answer:
Correct Answer: D
Solution:
The point of intersection is $ x=a\cos \theta +b\sin \theta $
$ y=a\sin \theta -b\cos \theta $ . Therefore, $ x^{2}+y^{2}=a^{2}+b^{2} $ . Obviously, it is equation of a circle with radius $\sqrt{a^{2}+b^{2}}$.
BETA
