1 Marks Question

Question 11 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 21 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 31 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 41 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 51 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 61 Mark

Write the equations of directrix of a hyperbola

Answer

$x= \pm \frac{a}{e}$

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Question 71 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 81 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 91 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 101 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 111 Mark

Write the equation of parabola having vertex at $(0,0)$ and focus at $(0,-a)$.

Answer

$x^2=-4 a y$

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Question 121 Mark

Define focal chord of the parabola.

Answer

Chord of the parabola which passes through foci is called focal chord.

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Question 131 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 141 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 151 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 161 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 171 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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