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Important Questions Class 11 Mathematics Chapter 11
Important Questions for CBSE Class 11 Mathematics Chapter 11 – Conic Sections
Conic Sections is Chapter 11 of the NCERT Book for Mathematics for Class 11. It is primarily concerned with various shapes, cones, and conic structures such as parabolas, hyperbolas, circles, axes, and the various subtopics associated with these. The following sections explain degenerate conic sections, standard equations of a parabola, Ellipse, Latus Rectum, and so on.
This is an important chapter that must be learned thoroughly through logical reasoning and practice. Students are advised to solve the Important Questions of this chapter, as each question is based on a different aspect and concept. Moreover, practising will improve problemsolving accuracy.
These topics are covered extensively in the CBSE examination. This chapter is also important in terms of competitive exams such as JEE Mains and JEE Advance. Students must master this chapter to excel in both school and competitive exams.
Extramarks Important Questions for Class 11 Mathematics Chapter 11 come with precise answers compiled by subject matter experts referring to the NCERT Books and past years’ question papers. Students can view the complete set of questions by clicking on the link provided here.
CBSE Class 11 Mathematics Chapter11 Important Questions
(add important questions)
Access NCERT Solutions for Mathematics Class 11 – Chapter 11 – Conic Sections
Given below is a set of Chapter 11 Class 11 Maths Important Questions. 1mark, 4marks, and 6mark answers and questions are discussed here.
Very Short Answer Questions: (1 Mark)
Q1. Find the length of the latus rectum of 2×2+3y2=18.
(a) 2 units
(b) 3 units
(c) 4 units
(d) None of these
A1. (c) 4 units
Q2. Find the length of the minor axis of x2+4y2=100.
(a) 10 units
(b) 12 units
(c) 14 units
(d) 8 units
A2. (a) 10 units
Q3. Find the equation of a circle with centre (b, a) & touching the xaxis.
(a) x2+y2−2bx+2ay+b2=0
(b) x2+y2+2bx−2ay+b2=0
(c) x2+y2−2bx−2ay+b2=0
(d) None of these
A3. (c) x2+y2−2bx−2ay+b2=0
Q4. Find the equations of the directrix & the axis of the parabola ⇒3×2=8y
(a) 3y−4=0, x=0
(b) 3x−2=0, X=0
(c) 3y−4x=0
(d) None of these
A4. (a) 3y−4=0, x=0
Long Answer Questions: (4 Marks)
Q1. Show that the equation x2+y2−6x+4y−36=0 represents a circle and find its centre & radius.
A1. It is of the form x2+y2+2gx+2Fy+c=0,
Where 2g=−6, 2f=4& c=−36
∴g=−3, f=2& c=−36
Thus, the centre of the circle is (−g,−f)=(3,−2)
The radius of the circle is g2+f2−c−−−−−−−−−√=9+4+36−−−−−−−−√
=7 units
Q2. Find the equation for the diameter of a circle drawn on the diagonal of a rectangle with sides x=6, x=3, y=3, and y=1.
A2. Let ABCD be the given rectangle and AD=x=−3, BC=x=6, AB=y=−1 & CD=y=−3.
Then A(−3,−1) and C(6,3).
The equation of the circle with AC as the diameter is:
(x+3)(x−6)+(y+1)(y−3)=0
⇒x2+y2−3x−2y−21=0
Q3. Show that the equation 6×2+6y2+24x−36y−18=0 represents a circle and find its centre & radius.
A3. 6×2+6y2+24x−36y−18=0
So, x2+y2+4x−6y+3=0
Where, 2g=4, 2f=−6 & c=3
∴g=2, f=−3 & c=3
Thus, the centre of the circle is (−g,−f)=(−2,3)
The radius of the circle =4+9+9−−−−−−−√=20−−√
=25–√ units
Q4. Find the equation for a circle whose endpoints for one of its diameters are A(2,3)& B(3,5).
A4. Let the endpoints of one of whose diameters are (x1,y1) and (x2,y2) be given by
(x−x1)(x−x2)+(y−y1)(y−y2)=0
So, x1=2, y1=−3 & x2=−3, y2=5.
∴ The required equation of the circle is (x−2)(x+3)+(y+3)(y−5)=0
⇒x2+y2+x−2y−21=0
Q5. Determine the focus and vertex coordinates, the equations of the directrix and the axis, and the length of the latus rectum of the parabola x2=8y.
A5. x2=−8y & x2=−4ay
So, 4a=8
⇒a=2
So it is the downward parabola.
Foci are F(0,−a) i.e. F(0,−2).
Vertex is O(0,0).
So, y=a=2.
Its axis is y− axis, whose equation is given by x=0.
Length of latus rectum=4a units.
=4×2 units
=8 units
Very Long Answer Questions: (6 Marks)
Q1. A man running on a race track notices that the distance between the two flag posts from him is always 12 m, and the distance between the flag posts is 10 m. Determine the equation of the man’s path.
A1. An ellipse is the locus of a point that moves so that the sum of its distances from two fixed points remains constant.
As a result, the path is an ellipse.
Let the ellipse equation be x2/a2+y2/b2=1.
Where b2=a2(1−c2)
It is clear that 2a=12 & 2ae=10
⇒a=b and e=56
⇒b2=a2(1−e2)
⇒b2=36(1−2536)
⇒b2=11
So, the required equation is x2/36+y2/11=1.
Q2. Determine the axis lengths, vertices coordinates, and foci. the eccentricity and length of the hyperbola latus rectum 25×2−9y2=225.
A2. 25×2−9y2=225
⇒x2/9−y2/25=1
Now, a2=9 & b2=25
And c=a2+b2−−−−−−√
⇒c=9+25−−−−−√
⇒c=34−−√
(i) Length of transverse axis =2a=2×3=6 units
Length of conjugate axis =2b=2×5=10 units
(ii) The coordinates of vertices are A(−a,0) & B(a,0) i.e. A(−3,0) & B(3,0)
(iii) The coordinates of foci are F1(−c,0) & F2(c,0) i.e. F1(−34−−√,0) & F2(34−−√,0)
(iv) Eccentricity, e=ca=34−−√3
(v) Length of the latus rectum =2b2a=503 units
Q3. Find the equation of the curve formed by the set of all these points, the sum of whose distances from A(4,0,0) and B(4,0,0) is 10 units.
A3. Let P(x,y,z) be an arbitrary point on the given curve.
So, PA+PB=10
⇒(x−4)2+y2+z2−−−−−−−−−−−−−−√+(x+4)2+y2+z2−−−−−−−−−−−−−−√=10
=(x+4)2+y2+z2−−−−−−−−−−−−−−√=10−(x−4)2+y2+z2−−−−−−−−−−−−−−√
Squaring both sides:
⇒(x+4)2+y2+z2=100+(x−4)2+y2+z2−20(x−4)2+y2+z2−−−−−−−−−−−−−−√
⇒16x=100−20(x−4)2+y2+z2−−−−−−−−−−−−−−√
⇒5(x−4)2+y2+z2−−−−−−−−−−−−−−√=25−4x
⇒25[(x−4)2+y2+z2]=625+16×2−200x
⇒9×2+25y2+25z2−225=0
So, the required equation of the curve is 9×2+25×2+25z2−225=0.
Q4. An equilateral triangle is inscribed in the parabola y2=4ax so that one of the triangle’s angular points is at the parabola’s vertex. Determine the length of each triangle side.
A4. Assume OPQ is an equilateral triangle inscribed in the parabola y2=4ax with the vertex O(0,0), so ∠POM=∠QOM=30∘ .
Let OP=OQ=r.
And P=(rcos30∘,rsin30∘)
⇒P=(r3–√2,r2)
P lies on the parabola.
⇒r24=4a(r3–√2)
⇒r=8a3–√
Hence, the length of each side of the triangle is 8a3–√ units.
Q.1 If the equation of a hyperbola is 9x^{2} ? 4y^{2} = 36, then the length of the latus rectum is
Marks:1
Ans
Q.2 If the equations of two diameters of a circle are xy = 6 and 2x + y = 6 and the radius of the circle is 6, then the equation of the circle is
Marks:1
Ans
x^{2} + y^{2}8x + 4y = 16.
Let the diameters of the circle be AB and CD, whose equations are
x y = 6 …(i) and 2x + y = 6 …(ii) respectively.
Solving (i) and (ii), we get: x = 4 and y = 2.
Since the point of intersection of any two diameters of a circle is its centre. Therefore, the coordinates of the centre of the required circle are (4, 2) and its radius is given as 6.
Hence, its equation is
(x4)^{2} + (y + 2)^{2} = 6^{2}
or
x^{2} + 168x + y^{2} + 4 + 4y = 36
x^{2} + y^{2}8x + 4y = 16.
Q.3
$\mathbf{\text{Find the equation of the hyperbola with foci}}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\pm}\mathbf{4}\mathbf{)}\mathbf{\text{and vertices}}$ $\mathbf{\text{}}(0,\pm \frac{\sqrt{13}}{2})\mathbf{.}$
Marks:4
Ans
$\begin{array}{l}\text{Since the foci is on}\mathrm{y}\text{axis, the equation of the hyperbola is of the form,}\\ \frac{{\mathrm{y}}^{2}}{{\mathrm{a}}^{2}}\frac{{\mathrm{x}}^{2}}{{\mathrm{b}}^{2}}=1.\\ \text{The vertices are}\left(0,\pm \frac{\sqrt{13}}{2}\right)\text{.}\\ \text{So},\mathrm{a}=\frac{\sqrt{13}}{2}\\ \text{Also, since the foci are}(0,\pm 4);\mathrm{c}=4\\ \mathrm{and}\text{}{\mathrm{b}}^{2}={\mathrm{c}}^{2}{\mathrm{a}}^{2}\\ ={4}^{2}{\left(\pm \frac{\sqrt{13}}{2}\right)}^{2}\\ =\frac{51}{4}\\ \text{Therefore, the equation of the hyperbola is}\\ \frac{{\mathrm{y}}^{2}}{\frac{13}{4}}\frac{{\mathrm{x}}^{2}}{\frac{51}{4}}=1\\ \text{i.e.,}204{\mathrm{y}}^{2}52{\mathrm{x}}^{2}=663.\end{array}$
Q.4 Find the equation of the parabola with focus (2, 0) and directrix x = ?2.
Marks:2
Ans
Since the focus lies on the xaxis, thus xaxis it self is the axis of the parabola.
Since the directrix is x = 2 and the focus is (2, 0), the parabola is to be of the form
y^{2} = 4ax with a = 2.
Hence the required equation is : y^{2} = 4(2)x = 8x.
Q.5 Find the equation of the circle whose centre lies at point (2,2) and passes through the centre of the circle x^{2} + y^{2} + 4x + 6y + 2 = 0
Marks:1
Ans
$\begin{array}{l}\text{The centre of the given circle is given by}(\mathrm{g},\mathrm{f})=(2,3)\\ \text{Radius of the required circle = distance between points}(2,2)\text{and}(2,3)\\ \sqrt{{\left({\mathrm{x}}_{1}{\mathrm{x}}_{2}\right)}^{2}{\left({\mathrm{y}}_{1}{\mathrm{y}}_{2}\right)}^{2}}=\sqrt{{\left(2\right(2\left)\right)}^{2}{2\left(3\right))}^{2}}\\ =\sqrt{16+25}=\sqrt{41}\\ \text{The equation of the circle whose centre lies at point}(2,2)\text{and passes through the}(2,3)\\ \text{is given by}\\ {\left(\mathrm{x}2\right)}^{2}+{\left(\mathrm{y}2\right)}^{2}={\left(\sqrt{41}\right)}^{2}\\ {\mathrm{x}}^{2}+{\mathrm{y}}^{2}4\mathrm{x}4\mathrm{y}33=0\end{array}$
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FAQs (Frequently Asked Questions)
1. What is an ellipse?
An ellipse is a point on a plane that moves in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed straight line remains constant and is always less than one. The major axis is defined as the line segment connecting the ellipse’s foci to the ellipse’s endpoint. Similarly, the minor axis is defined as the line segment passing through the centre and being perpendicular to the major axis, with endpoints on the ellipse.
2. Define the horizontal, vertical, and special forms of the ellipse.
Consider the following equation: x2/a2+y2/b2=1, 0 b a.
A horizontal ellipse is the condition in which the coefficient of x2 has a larger denominator than the major axis, which lies along the xaxis.
A vertical ellipse is defined as the condition where the denominator of the coefficient of x 2 is less than the major axis along the yaxis.
When the equation is written in the special form of an ellipse (h,k) and the axes are parallel to the coordinate axes, the equation is (xh)2/a2+(yk)2/b2=1.
3. What is the Latus rectum and what does it mean in a parabola and an ellipse?
A line segment that is perpendicular to a given axis is defined as a latus rectum. The Latus rectum in a parabola is the line segment that is perpendicular to the parabola’s axis and passes through the focus. Its ends are on the parabola. The Latus rectum in an ellipse is a line segment that runs perpendicular to the main axis through any of the foci and has endpoints on the ellipse.
4. How are curves defined on a coordinate plane?
An equation is used to track and plot the behaviour of a point on a coordinate plane. These equations are used to represent conic sections. Every point on the curves must adhere to the equation.