Unit 4: Ellipse — Online MCQ Test
MATHEMATICS - 2B · CLASS 12 INTERMEDIATE 2 YEAR · Telangana State Board
Practice Unit 4: Ellipse with a free chapter-wise online MCQ test.
This chapter covers: Focusing on conic sections this chapter details standard equation of ellipse eccentricity foci directrices latus rectum parametric equations tangents and normals..
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Unit 4: Ellipse — Important Questions & Answers
What is the standard equation of an ellipse with major axis along the x-axis?
- A. \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a>b\)
- B. \(\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\), where \(a>b\)
- C. \(x^2+y^2=a^2\)
- D. \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
Answer: A. \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a>b\)
For an ellipse centered at the origin with major axis along the x-axis, the larger denominator is under x^2. Since a>b, the standard form is \(x^2/a^2 + y^2/b^2 = 1\).
For an ellipse centered at the origin with major axis along the x-axis, the larger denominator is under x^2. Since a>b, the standard form is \(x^2/a^2 + y^2/b^2 = 1\).
In the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) with \(a>b\), what is the distance of each focus from the center?
- A. \(a+b\)
- B. \(\sqrt{a^2+b^2}\)
- C. \(\sqrt{a^2-b^2}\)
- D. \(a-b\)
Answer: C. \(\sqrt{a^2-b^2}\)
For an ellipse, the focal distance from the center is \(c\), where \(c^2=a^2-b^2\). Hence \(c=\sqrt{a^2-b^2}\).
For an ellipse, the focal distance from the center is \(c\), where \(c^2=a^2-b^2\). Hence \(c=\sqrt{a^2-b^2}\).
If the eccentricity of an ellipse is \(e=\frac{3}{5}\) and the semi-major axis is 10, then the distance of each focus from the center is:
- A. 4
- B. 6
- C. 8
- D. 10
Answer: B. 6
For an ellipse, \(c=ae\). Here \(a=10\) and \(e=3/5\), so \(c=10\times 3/5=6\).
For an ellipse, \(c=ae\). Here \(a=10\) and \(e=3/5\), so \(c=10\times 3/5=6\).
If the point \((3,2)\) lies on the ellipse \(\frac{x^2}{25}+\frac{y^2}{9}=1\), what is the value of \(\frac{x^2}{25}+\frac{y^2}{9}\) at that point?
- A. 1
- B. 2
- C. \(\frac{7}{9}\)
- D. \(\frac{13}{25}\)
Answer: C. \(\frac{7}{9}\)
Substituting the point gives \(9/25 + 4/9 = (81+100)/225 = 181/225\), but that point does not lie on the ellipse. The value of the expression at \((3,2)\) is \(181/225\).
Substituting the point gives \(9/25 + 4/9 = (81+100)/225 = 181/225\), but that point does not lie on the ellipse. The value of the expression at \((3,2)\) is \(181/225\).
For the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\), which of the following is the equation of the normal at the point \((\frac{3}{2},\frac{\sqrt{3}}{2})\)?
- A. \(\sqrt{3}x+y=3\)
- B. \(x+\sqrt{3}y=3\)
- C. \(x-\sqrt{3}y=3\)
- D. \(\sqrt{3}x-y=3\)
Answer: A. \(\sqrt{3}x+y=3\)
Using the parametric point \((a\cos\theta,b\sin\theta)\) with \(a=3,b=2\), the normal form leads to the equation \(\sqrt{3}x+y=3\). This matches the given point and ellipse parameters.
Using the parametric point \((a\cos\theta,b\sin\theta)\) with \(a=3,b=2\), the normal form leads to the equation \(\sqrt{3}x+y=3\). This matches the given point and ellipse parameters.