Equation of the Circle Using End Points of Diameter

Q.

The equation of the circumcircle of an equilateral triangle is $x^2+y^2+2gx+2fy+c=0$ and one vertex of the triangle is $\left(1,1\right)$ . The equation of incircle of the triangle is

1. $4\left(x^2+y^2\right)=g^2+f^2$

2. $4\left(x^2+y^2\right)+8gx+8fy=\left(1-g\right)\left(1+3g\right)+\left(1-f\right)\left(1+3f\right)$

3. $4\left(x^2+y^2\right)+8gx+8fy=g^2+f^2$

4. None of these

Q. Equation of the circle passing through $(4,3)$ and centre at the origin is _______ .

1. $(x-4{)}^2+(y-3{)}^2=7$
2. $x^2+y^2=25$
3. $x^2+y^2=49$
4. $(x-3{)}^2+(y-4{)}^2=25$

Q.

Find the coordinates of the centre of the circle passing through the points (0, 0), (–2, 1) and (–3, 2). Also, find its radius. [4 MARKS]

Q.

The radius of the circle $x^2+y^2-2x+4y-11=0$
is .

1. -1 units
2. 2 units
3. 11 units
4. 4 units

Q.

Consider the points $A(3,2),B(9,4),C(7,10)$. Then the circle with AC as diameter doesn't pass through B.

1. False

2. True

Q. Find the radius of the circle $x^2+y^2-2x+4y-11=0$.

1. 4

Q.

The equation of the circle having $(-2,1)$ and $(4,-1)$ as the endpoints of its diameter is

1. $x^2+y^2-2x-9=0$

2. $x^2+y^2-2x+9=0$

3. $x^2+y^2+2x+9=0$

4. $x^2+y^2+2x-9=0$

Q.

Consider the points $A(4,3)$ and $B(0,1)$. If a circle is drawn with AB as diameter, then the coordinates of the points where this circle intersects the $x$-axis are

1. $(-3,0)$ and $(2,0)$.

2. $(3,0)$ and $(-1,0)$.

3. $(3,0)$ and $(1,0)$

4. $(2,0)$ and $(-1,0)$.

Q.

A circle is drawn with AB as the diameter whose endpoints are $A(2,4)$ and $B(4,12)$. If another circle with diameter one-third of the above circle is drawn with the same centre, what are the points that the circle cuts AB?

1. $(6,\frac{28}{3})$ and $(6,\frac{20}{3})$

2. $(10,\frac{28}{3})$ and $(6,\frac{20}{3})$

3. $(10,\frac{28}{3})$ and $(10,\frac{20}{3})$

4. $(10,\frac{20}{3})$ and $(6,\frac{20}{3})$

Q.

The equation of the circle having $(-2,-1)$ and $(2,1)$ as the endpoints of its diameter is

1. $x^2+y^2=5$

2. $x^2-y^2=5$

3. $x^2+y^2=-5$

4. $x^2-y^2=-5$

Q. Question 14
If the centre of a circle is (2a, a-7), then find the values of a , if the circle passes through the point (11, -9) and has diameter $10\sqrt{2}$ units.

Q.

The equation of circle with centre $(1,1)$ and radius 4 is given by $(x-1{)}^2+(y-1{)}^2=4$

1. True

2. False

Q.

A circle is drawn with AB as the diameter whose endpoints are $A(2,4)$ and $B(4,12)$. If another circle with diameter one-third of the above circle is drawn with the same centre, what are the points that the circle cuts AB?

1. $(6,\frac{28}{3})$ and $(6,\frac{20}{3})$

2. $(10,\frac{28}{3})$ and $(6,\frac{20}{3})$

3. $(10,\frac{28}{3})$ and $(10,\frac{20}{3})$

4. $(10,\frac{20}{3})$ and $(6,\frac{20}{3})$

Q. The Class $X$ students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Sapling of Gulmoharare planted on the boundary at a distance of $1$ m from each other. There is a triangular grassy lawn in the plot as shown in the Fig. The students are to sow seeds of flowering plants on the remaining area of the plot.
(i) Taking $A$ as origin, find the coordinates of the vertices of the triangle.
(ii) What will be the coordinates of the vertices of $\Delta PQR$ if $C$ is the origin?
Also calculate the areas of the triangles in these cases. What do you observe?

Q.

The equation of the circle having $(3,3)$ and origin as the endpoints of its diameter is

1. $x^2+y^2-3(x+y)=0$

2. $x^2+y^2-3x+y=0$

3. $(x+y{)}^2-2xy-3(x+y)=0$

4. $(x+y{)}^2+2xy-3x+y=0$

Q. Find the center of the circle given by the equation $x^2+y^2-4x+6y-51=0$.

1. (3, -2)
2. (-3, 2)
3. (2, -3)
4. (-2, 3)

Q. How are the points $(-2,2),(2,3)$ and $(2,-3)$ situated with respect to the circle $x^2+y^2-4x–2y+3=0$

Q. $A(-3,9)$ and $B(9,-1)$ are the endpoints of the diameter of a circle. The equation for this circle is

1. $(x-3{)}^2+(y-4{)}^2=61$
2. $(x-7{)}^2+(y+4{)}^2=269$
3. $(x+7{)}^2+(y-4{)}^2=61$
4. $(x+3{)}^2+(y+4{)}^2=169$
5. $(x+3{)}^2+(y-4{)}^2=25$

Q. If the origin is shifted to the point (1, 1), axes remaining parallel, find the new equation of the locus in each of the following.
i) $xy-x-y+1=0$
ii) $x^2-y^2-2x+2y=0$
iii) $x^2+y^2-4x+6y+3=0$

Q.

The equation of the circle having $(6,7)$ and $(4,3)$ as the endpoints of its diameter is

1. $(x+y{)}^2-2xy+45=0$

2. $(x^2-y^2-2xy+45=0$

3. $10(x+y)-2xy+45=0$

4. $(x+y)\left[\right(x+y)-10]-2xy+45=0$

Q.

Equation of the circle passing through $(2,3)$ and centre at the origin is _______ .

1. $(x-2{)}^2+(y-3{)}^2=13$

2. $x^2+y^2=13$

3. $(x-3{)}^2+(y-2{)}^2=13$

4. $x^2-y^2=13$

Q.

The equation of the circle having $(4,-1)$ and $(2,-1)$ as the endpoints of its diameter is

1. $x^2+y^2-x+2y+9=0$

2. $x^2+y^2-6x+2y-9=0$

3. $x^2+y^2+x+y-9=0$

4. $x^2+y^2-6x+2y+9=0$

Q.

Find the radius of the circle $x^2+y^2-2x+4y-11=0$

___

Q. Equation of the circle passing through $(2,3)$ and centre at the origin is .

1. $x^2+y^2=13$
2. $x^2+y^2=12$
3. $x^2+y^2=11$
4. $x^2+y^2=1$

Q. Find the equation of the circle when the end points of a diameter are $A(3,2)$ and $B(3,5)$

Q. Compare the given equation of the circle and then find the value of $a+b+c+d+e$ if the centre of the circle is at origin (2, 0) and radius is 6.
given equation : $a{x}^2+b{y}^2+cx+dy+e=0$

Q. If $(6,-3)$ is the one extremity of diameter to the circle$x^2+y^2-3x+8y-3=0$ then its other extremity is -

1. $(3,-5)$
2. $(\frac{3}{2},-4)$
3. $(-3,-5)$
4. $(3,5)$

Q.

The equation of the circle having $(4,2)$ and $(2,-12)$ as the end points of its diameter is

1. $x^2+y^2+6x+10y+16=0$

2. $x^2+y^2-6x+10y-16=0$

3. $x^2+y^2+10x-6y+16=0$

4. $x^2+y^2-10x+6y-16=0$

Q. Find the center of the circle given by the equation $x^2+y^2-4x+6y-51=0$.

Q. Find the centre and the radius of the circle which is concentric to the circle $x^2+y^2-16=0$ having radius double of the radius of the given circle.