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Question

The equation2y2-xy-x2+6x-8=0 represents


A

A pair of straight lines

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B

A circle

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C

An ellipse

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D

A parabola

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Solution

The correct option is C

An ellipse


Identifying the curve:

Given 2y2-xy-x2+6x-8=0

The standard form of the equation is ax2+2hxy+by2+2gx+2fy+c=0

Comparing the above equation with the given equation, we get

a=-1,h=−12​,b=2,g=3,f=0​,c=-8

We know that the condition for straight lines is given by
Δ=abc+2fgh−af2−bg2−ch2

Substituting the values, we get

=(-1)(2)(-8)+20(3)-12−(-1)02−(2)(3)2−(-8)-122=16+0+0−18+2​⇒0=0

Discriminant =b2−4ac

=(4​)2−4(-1)(-8)=16​−32=−16<0

We know that if b2−4ac<0, then the curve represents either a circle or an ellipse.

Now from the values, we know a≠c.

For a curve to be an ellipse, b2−4ac<0 and a≠c. This proves that the given curve is an ellipse.

Hence, option (C) is the correct answer.


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