MCQ

MCQ 1512 Marks

A circle which passes through origin and cuts intercepts on axes a and b, the equation of circle is

✓
$x^2+y^2-a x-b y=0$
B
$x^2+y^2+a x+b y=0$
C
$x^2+y^2-a x+b y=0$
D
$x^2+y^2+a x-b y=0$

Answer

Correct option: A.

$x^2+y^2-a x-b y=0$

(A)
Centre is $\left(\frac{ a }{2}, \frac{b}{2}\right)$ and radius $=\sqrt{\frac{ a ^2+ b ^2}{4}}$

Hence, equation of circle is
$x^2+y^2-a x-b y=0$

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MCQ 1522 Marks

If $(\alpha, \beta)$ is the centre of a circle passing through the origin, then its equation is

A
$x^2+y^2-\alpha x-\beta y=0$
B
$x^2+y^2+2 \alpha x+2 \beta y=0$
✓
$x^2+y^2-2 \alpha x-2 \beta y=0$
D
$x^2+y^2+\alpha x+\beta y=0$

Answer

Correct option: C.

$x^2+y^2-2 \alpha x-2 \beta y=0$

(C)
Radius $=$ Distance from origin $=\sqrt{\alpha^2+\beta^2}$
$\therefore(x-\alpha)^2+(y-\beta)^2=\alpha^2+\beta^2$
$\Rightarrow x^2+y^2-2 \alpha x-2 \beta y=0$

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MCQ 1532 Marks

A circle touches the Y-axis at the point (0, 4) and cuts the X-axis in a chord of length 6 units. The radius of the circle is

A
3
B
4
✓
5
D
6

Answer

Correct option: C.

From the figure,
Radius $(r)=\sqrt{(4)^2+(3)^2}=5$

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MCQ 1542 Marks

Equation of circle with centre $(- a,- b)$ and radius $\sqrt{a^2-b^2}$ is

✓
$x^2+y^2+2 a x+2 b y+2 b^2=0$
B
$x^2+y^2-2 a x-2 b y-2 b^2=0$
C
$x^2+y^2-2 a x-2 b y+2 b^2=0$
D
$x^2+y^2-2 a x+2 b y+2 a ^2=0$

Answer

Correct option: A.

$x^2+y^2+2 a x+2 b y+2 b^2=0$

(A)
$x^2+y^2+2 a x+2 b y+2 b^2=0$
Centre $=(-a,-b)$
∴ option (A) is the correct answer.

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MCQ 1552 Marks

The equation of the circle which touches X-axis and whose centre is (1, 2), is

A
$x^2+y^2-2 x+4 y+1=0$
✓
$x^2+y^2-2 x-4 y+1=0$
C
$x^2+y^2+2 x+4 y+1=0$
D
$x^2+y^2+4 x+2 y+4=0$

Answer

Correct option: B.

$x^2+y^2-2 x-4 y+1=0$

(B)
Since the circle touches X -axis,
radius $=2$.
$\therefore \quad$ the equation of the circle is
$(x-1)^2+(y-2)^2=2^2$
$\Rightarrow x^2+y^2-2 x-4 y+1=0$

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MCQ 1562 Marks

The equation of the circle which touches both axes and whose centre is (x1, y1) is

A
$x^2+y^2+2 x_1(x+y)+x_1^2=0$
✓
$x^2+y^2-2 x_1(x+y)+x_1^2=0$
C
$x^2+y^2=x_1^2+y_1^2$
D
$x^2+y^2+2 x x_1+2 y y_1=0$

Answer

Correct option: B.

$x^2+y^2-2 x_1(x+y)+x_1^2=0$

(B)
The equation of circle with centre $\left(x_1, y_1\right)$ is
$\left(x-x_1\right)^2+\left(y-y_1\right)^2=r^2$
Since the circle touches both the axes,
$x_1=y_1= r$
$\therefore\left(x-x_1\right)^2+\left(y-x_1\right)^2=x_1^2$
$\Rightarrow x^2+y^2-2 x_1(x+y)+x_1^2=0$

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MCQ 1572 Marks

The equation of the circle which touches both the axes and whose radius is a, is

✓
$x^2+y^2-2 a x-2 a y+a^2=0$
B
$x^2+y^2+a x+a y-a^2=0$
C
$x^2+y^2+2 a x+2 a y- a ^2=0$
D
$x^2+y^2- a x- a y+ a ^2=0$

Answer

Correct option: A.

$x^2+y^2-2 a x-2 a y+a^2=0$

(A)
Required equation is $(x-a)^2+(y-a)^2=a^2$
$\Rightarrow x^2+y^2-2 a x-2 a y+a^2=0$

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MCQ 1582 Marks

Centre of the circle $(x-3)^2+(y-4)^2=5$ is

✓
(3, 4)
B
(-3, -4)
C
(4, 3)
D
(-4, -3)

Answer

Correct option: A.

(3, 4)

(A)

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Maths MCQ