Conic Sections Question 515
Question: The equation $ x^{2}-2xy+y^{2}+3x+2=0 $ represents
[UPSEAT 2001]
Options:
A) 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 upwards is $ y = ax^2 + bx + c $, where $ a $ determines the width and direction of the parabola.
B) 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 (foci) is constant.
C) A hyperbola is a conic section formed by the intersection of a plane with a double-napped cone, where the plane intersects both nappes. It consists of two separate branches and has two asymptotes.
D) A circle
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Answer:
Correct Answer: A
Solution:
$ \Delta =(1)(1)(2)+2( \frac{3}{2} )(0)(-1)-(1){{(0)}^{2}} $
$ (1){{( \frac{3}{2} )}^{2}}-2{{(-1)}^{2}} $
$ =2-\frac{9}{4}-2<0 $ and $ h^{2}-ab=1-1=0 $ . i.e., $ h^{2}=ab $
therefore, a parabola.
BETA
