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Question

Let C be the curve(s) given by (ax2+by2+c)(ay2bx)=0;a,b,cR{0} and ab. Then C represents

A
a parabola only if a,b and c are of the same sign.
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B
a circle or a parabola if a and b are of the same sign and c is of opposite sign.
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C
an ellipse or a parabola if a and b are of the same sign and c is of opposite sign.
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D
a hyperbola or a parabola if a and b are of opposite sign
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Solution

The correct option is D a hyperbola or a parabola if a and b are of opposite sign
(ax2+by2+c)(ay2bx)=0
ax2+by2+c=0 or ay2bx=0
Here ay2bx=0 represents a parabola a,bR{0}

(i) For ax2+by2+c=0
If a,b,c are of the same sign, then ax2+by2+c=0 is not possible. (a,b,cR{0})
So, C will represent parabola only

(ii) For ax2+by2=c,
x2(c/a)+y2(c/b)=1
For the above equation to represent an Ellipse,
(ca)>0,(cb)>0
(ca)<0,(cb)<0
a and b should be of the same sign but c should be of opposite sign.

(iii)For ax2+by2=c to represent hyperbola, (ca)(cb)<0
ab<0
a and b should be of the opposite sign

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