The equation $3 x^{2} + 10 x y + 11 y^{2} + 14 x + 12 y + 5 = 0$ represents

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — an ellipse

Detailed Solution

Compare with $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ : $a = 3$ , $h = 5$ , $b = 11$ , $g = 7$ , $f = 6$ , $c = 5$ .

$Δ = ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 165 + 420 - 108 - 539 - 125 = - 187 \neq = 0$ , so the conic is not a pair of lines.

$h^{2} - ab = 25 - 33 = - 8 < 0$ , so it is an ellipse.

✓ Check

Putting

y = 0

gives

3 x^{2} + 14 x + 5 = 0

with discriminant

196 - 60 = 136 > 0

, so the curve has real points.

⚠ Common trap

(A) a circle needs

a = b

and

h = 0

, but here

3 \neq = 11

and

h = 5

Quick tip

$h^{2} < ab$ : ellipse; $h^{2} = ab$ : parabola; $h^{2} > ab$ : hyperbola (with $Δ \neq = 0$ )

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Conic Sections — Preparation

This question tests your understanding of Conic Sections, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

Conic sections are the curves formed when a plane cuts a double cone: the circle, parabola, ellipse and hyperbola. They can all be defined using a focus, a directrix and the eccentricity e, which is the ratio of a point's distance from the focus to its distance from the directrix. A conic is a parabola when e = 1, an ellipse when e < 1, and a hyperbola when e > 1. This topic brings the three conics together and tests identifying a conic from its equation, comparing their standard forms, and finding the focus, directrix, eccentricity and latus rectum. Mixed questions often ask you to recognise which conic an equation represents before applying the right formulas. For in-depth practice on each curve, see the separate parabola, ellipse and hyperbola pages. Practise the conic section questions below, each with a step-by-step solution.

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