2 Marks Questions

Question 12 Marks

Find the equation of the circle with centre $C(-2,3)$ and which touches the line $x-y+7=0$.

Answer

The given equation of line is $x-y+7=0\quad \ldots(i) $
Let $r$ be the radius of required circle, then
$r=$ perpendicular distance of $(-2,3)$ from the line $( i )$
$\begin{array}{l}=\frac{|-2-3+7|}{\sqrt{(1)^2+(-1)^2}} \\ =\frac{2}{\sqrt{2}} \\ =\sqrt{2}\end{array}$
$\therefore$ The equation of the required circle is
$(x+2)^2+(y-3)^2=(\sqrt{2})^2$
or, $x^2+y^2+4 x-6 y+11=0$.

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Question 22 Marks

If a circle passes through the points $(0,0),(a, 0)$ and $(0, b)$, then find the coordinates of its centre.

Answer

The circle passes through the points $O(0,0), A(a, 0)$ and $B(0, b)$.
As the axes are perpendicular to each other, then $\angle A O B=90^{\circ}$ and hence $A B$ becomes a diameter of the circle. The centre of the circle is the mid-point of the segment $A B$.

$\therefore$ The centre of circle is $\left(\frac{a+0}{2}, \frac{0+b}{2}\right)$ i.e., $\left(\frac{a}{2}, \frac{b}{2}\right)$.

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Question 32 Marks

A circle of radius $r$ is in the second quadrant. If the circle touches both the axes, then find the equation of the circle.

Answer

Since, the circle is in the second quadrant, its centre will also be in the second quadrant. Also, the circle touches the axes, so that coordinates of its centre will be $(-r, r)$.

Hence, the required equation of the circle is
$
\begin{array}{l}
(x+r)^2+(y-r)^2=r^2 \\
\text { or, }\left(x^2+2 r x+r^2\right)+\left(y^2-2 r y+r^2\right)=r^2 \\
\text { or } x^2+y^2+2 r x-2 r y+r^2=0 \text {. }
\end{array}
$

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Question 42 Marks

Find the parametric equations of the circle $x^2+y^2-2 x+4 y-4=0$.

Answer

We know that, the parametric equations of the circle $(x-h)^2+(y-k)^2=r^2$ are
$x=h+r \cos t$ and $y=k+r \sin t, 0 \leq t \leq 2 \pi$
The given equation of circle is
$x^2+y^2-2 x+4 y-4=0$
It can be written as
$\left(x^2-2 x+1\right)+\left(y^2+4 y+4\right)=1+4+4$
or, $(x-1)^2+(y+2)^2=3^2$, which is comparable with $(x-h)^2+(y-k)^2=r^2$.
Here, $h=1, k=-2$ and $r=3$
Therefore, parametric equations of the given circle are :
$x=1+3 \cos t \text { and } y=-2+3 \sin t, 0 \leq t \leq 2 \pi .$

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Applied Maths 2 Marks Questions

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