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MATHS 1 Marks Question

PART - 2 CH - 10 Conic Sections · Rajasthan Board (RBSE) STD 11 Science (Rajasthan - English Medium). 17 questions.

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1 Marks Question

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17 questions · timed · auto-graded

Question 11 Mark
Write the equation of hyperbola whose transverse axis and conjugate axis are 4 and 5 respectively.
Answer

$\begin{array}{l} \text { Given that : } \quad 2 a=4, \quad 2 b=5 \\ \Rightarrow \quad a=2 \quad \Rightarrow \quad b=\frac{5}{2} \\ \Rightarrow \$a)^2=4 \quad \text { and } b^2=\frac{25}{4} \end{array}$
So, equation of hyperbola :
$\begin{array}{lc} \Rightarrow & \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \\ \Rightarrow & \frac{x^2}{4}-\frac{y^2}{\frac{25}{4}}=1 \\ \Rightarrow & \frac{x^2}{4}-\frac{4 y^2}{25}=1 \\ \Rightarrow & 25 x^2-16 y^2=100 \end{array}$
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Question 21 Mark
Define point circle.
Answer
When the value of radius of a circle is zero, then equation of circle
$\quad$$\quad$$(x-h)^2+(y-k)^2=0$
In this way the circle becomes smaller and only one point remains, then it is called a point circle. $x^2+y^2$ $=0$ is a point circle at origin $(0,0)$.
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Question 31 Mark
Find the eccentricity of rectangular hyperbola.
Answer
We know that
$b^2=a^2\left(e^2-1\right)$
Here $b=a$
So, $a^2=a^2\left(e^2-1\right)$
$\Rightarrow \quad e^2=2 \Rightarrow e=\sqrt{2}$
So eccentricity of rectangular hyperbola is 2 .
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Question 51 Mark
Write the equation of directrix of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $(a>b)$.
Answer
$x= \pm \frac{a}{e}$
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Question 61 Mark
Write the formula of eccentricity of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
Answer
Eccentricity of ellipse $e=\sqrt{1-\frac{b^2}{a^2}}$
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Question 81 Mark
Define latus rectum of parabola. Write the value of its length.
Answer
Chord of the parabola which passes through foci and perpendicular to axis of parabola, is called latus rectum. For parabola $y^2=4 a x$, length of latus rectum is always $4 a$.
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Question 91 Mark
Write the centre and radius of the circle : \[(x+5)^2+(y+1)^2=9\]
Answer
On comparing the given circle with $(x-h)^2+(y-k)^2= R ^2$
$\Rightarrow h=-5, k=-1$ and $R =3$
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Question 101 Mark
Write the vertex of the hyperbola.
Answer
$\begin{array}{l}4 x^2-9 y^2=36 \\\Rightarrow \quad \frac{x^2}{9}-\frac{y^2}{4}=1\end{array}$
Vertex of hyperbola $( \pm a, 0)=( \pm 3,0)$
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Question 111 Mark
Equation $x^2+y^2+2 g x+2 f y+c=0$ always represents a circle and write its centre and radius.
Answer
Centre of circle $(-g,-f)$ and radius $r=$ $\sqrt{g^2+f^2-c}$
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Question 121 Mark
Find the equation of that circle whose centre is ( $a$ cos $\theta, a \sin \theta$ ) and radius is $a$.
Answer

$\begin{array}{l}(x-h)^2+(y-k)^2=\text { (radius) }^2 \\ \Rightarrow \quad(x-a \cos \theta)^2+(y-a \sin \theta)^2=a^2 \\ \Rightarrow \quad x^2-2 a \cos \theta \cdot x+a^2 \cos ^2 \theta+y^2-2 a \sin \theta . \\ \quad y+a^2 \sin ^2 \theta=a^2 \\ \Rightarrow \quad x^2+y^2-2 a \cos \theta x-2 a \sin \theta y+ \\ \quad a^2\left(\cos ^2 \theta+\sin ^2 \theta\right)=a^2 \\ \Rightarrow \quad x^2+y^2-2 a \cos \theta x-2 a \sin \theta y=0\end{array}$
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Question 131 Mark
Define hyperbola.
Answer
A hyperbola is the locus of a moving point such that the difference of its distances from two fixed points is always constant.
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Question 141 Mark
Write the coordinates of the ends of the latus rectum of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Answer
$\left(a e, \frac{b^2}{a}\right),\left(a e,-\frac{b^2}{a}\right)$
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Question 151 Mark
Write the value of eccentricity of hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Answer
Eccentricity $(e)=\sqrt{1+\frac{b^2}{a^2}}$
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Question 161 Mark
Define latus rectum of ellipse. Write its length.
Answer
The chord of the ellipse passing through focus and perpendicular to major axis of the ellipse is called latus rectum.
Its length is $\frac{2 b^2}{a}$.
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Question 171 Mark
Define focal chord of the parabola.
Answer
Chord of the parabola which passes through foci is called focal chord.
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