Find the number of point(s) of intersection of the ellipse x^24+(y-1)^29=1 and the circle x^2+y^2=4

B. 3. Step 1: From the circle, x^2=4-y^2. Substituting: 4-y^24+(y-1)^29=1, so 4(y-1)^2=9y^2 and 2(y-1)=± 3y. Step 2: This gives y=-2 or y=2/5. Step 3: y=-2 gives x^2=0, one point (0,-2) where the curves touch; y=2/5 gives x^2=96/25, two points. Total 3. Check: At (0,-2): ellipse 0+9/9=1 and circle 0+4=4; it is the lowest point of both curves, so they touch there. Trap: (A) 4 counts y=-2 as two points, but x^2=0 gives only x=0.

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NIMCET 2020

Find the number of point(s) of intersection of the ellipse $\frac{x ^{2}}{4} + \frac{( y - 1 ) ^{2}}{9} = 1$ and the circle $x^{2} + y^{2} = 4$

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Answer: Option B — 3

Detailed Solution

From the circle, $x^{2} = 4 - y^{2}$ . Substituting: $\frac{4 - y ^{2}}{4} + \frac{( y - 1 ) ^{2}}{9} = 1$ , so $4 (y - 1)^{2} = 9 y^{2}$ and $2 (y - 1) = \pm 3 y$ .

This gives $y = - 2$ or $y = \frac{2}{5}$ .

$y = - 2$ gives $x^{2} = 0$ , one point $(0, - 2)$ where the curves touch; $y = \frac{2}{5}$ gives $x^{2} = \frac{96}{25}$ , two points. Total 3.

✓ Check

(0, - 2)

: ellipse

0 + \frac{9}{9} = 1

and circle

0 + 4 = 4

; it is the lowest point of both curves, so they touch there.

⚠ Common trap

(A) 4 counts

y = - 2

as two points, but

x^{2} = 0

gives only

x = 0

Quick tip

$4 (y - 1)^{2} = 9 y^{2} \Rightarrow y = - 2, \frac{2}{5}$ : 1 + 2 = 3 points

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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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Find the number of point(s) of intersection of the ellipse $\frac{x ^{2}}{4} + \frac{( y - 1 ) ^{2}}{9} = 1$ and the circle $x^{2} + y^{2} = 4$

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Find the number of point(s) of intersection of the ellipse $\frac{x ^{2}}{4} + \frac{( y - 1 ) ^{2}}{9} = 1$ and the circle $x^{2} + y^{2} = 4$