Circles Test 20

Result

Circles Test 20

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Weekly Quiz Competition

Question 1
1 / -0

The equation of the circle which passes through the point (3,-2) and (-2,0) and centre line 2x-y=3,is

A
$${ x }^{ 2 }+{ y }^{ 2 }-3x-12y+2=0$$

B
$${ x }^{ 2 }+{ y }^{ 2 }-3x+12y+2=0$$

C
$${ x }^{ 2 }+{ y }^{ 2 }+3x+12y+2=0$$

D
None of these

Solution

Let $$x^2+y^2+2gx+2fy+c=0$$........(1) be the equation $$(3,-2)$$ lies on the circle.

Plugging in

$$13+6g-4f+c=0$$.........(2)

$$(-2,0)$$ lies on the circle lies on the circle

$$4-2g+c=0$$........(3)

centre $$(-g,-f)$$ lies on $$2x-y-3=0$$

$$-2g+f-3=0$$..........(4)

This (4) equation including the first and $$3$$ unknowns g,f, and c (Note $$x$$ and $$y$$ are considered constant) conditions for consistency given.

$$\begin{vmatrix} x^2+y^2& 2x& 2y& 1\\ 13& 6& -4& 1\\ 4& -4x& 0& 1\\ -3& -2& 1& 0\end{vmatrix}=0$$

After somehow operations and expanding (one can directly expand also)

$$\boxed{x^2+y^2+3x+12y+2=0}$$
Question 2
1 / -0

A circle has radius $$3$$ units and its centre lies on the line $$y=x-1$$. Then the equation of this circle if it passes through the point $$(7, 3)$$, is?

A
$$x^2+y^2-8x-6y=16$$

B
$$x^2+y^2+8x+6y+16=0$$

C
$$x^2+y^2-8x-6y-16=0$$

D
None of these

Solution
Question 3
1 / -0

The eq. of the circle which touches the exes of y at a distance of $$4$$ from the origin and cuts the intercepts of $$6$$ units from the axis of x is

A
$$x^2 +y^2 \pm 10x \pm 8y + 16=0$$

B
$$x^2 +y^2 \pm 5x \pm 4y + 16=0$$

C
$$x^2 +y^2 \pm 10x \pm 8y - 16=0$$

D
$$x^2 +y^2 \pm 5x \pm 4y - 16=0$$

Solution
Question 4
1 / -0

The equation of the circle having as a diameter, the chord $$x - y - 1 = 0$$ of the circle $$2x^2 + 2y^2 - 2x - 6y - 25 = 0$$, is

A
$$x^2 + y^2 - 3x - y - \dfrac{29}{2} = 0$$

B
$$2x^2 + 2y^2 + 2x - 5y - \dfrac{29}{2} = 0$$

C
$$2x^2 + 2y^2 - 6x - 2y - 21 = 0$$

D
None of these

Solution
Question 5
1 / -0

Let $$P,\ Q,\ R,\ S$$ be the feet of perpendicular drawn from the point $$(1,\ 1)$$ the lines $$y=3x+4$$ and $$y=-3x+6$$ and their angle bisectors respectively, the equation of the circle whose extremities of a diameter are $$R$$ and $$S$$ is

A
$$3x^{2}+3y^{2}+104x-110=0$$

B
$$x^{2}+y^{2}+104x-110=0$$

C
$$3x^{2}+3y^{2}-4x-18y+16=0$$

D
$$x^{2}+y^{2}-4x-18+16=0$$

Solution
Question 6
1 / -0

The intercept on the line $$y = x$$ by the circle $${x^2} + {y^2} - 2x = 0$$ is $$AB$$. The equation of the circle with $$AB$$ as a diameter, is

A
$${x^2} + {y^2} + x + y = 0$$

B
$${x^2} + {y^2} = x + y$$

C
$${x^2} + {y^2} - 3x + y = 0$$

D
None of the above

Solution

Hence, Option (B) is the correct answer.
Question 7
1 / -0

The equation of the circle passing through the points of intersection of the lines $$2x+y=0,\ x+y+3=0$$ and $$x-2y=0$$ is

A
$$x^{2}+y^{2}-x+7y=0$$

B
$$x^{2}+y^{2}+x-7y=0$$

C
$$x^{2}+y^{2}-x-7y=0$$

D
$$x^{2}+y^{2}+x+7y=0$$

Solution
Question 8
1 / -0

Let $$A B C$$ be a triangle, from vertices $$A , B$$ and $$C$$ altitudes $$A D , B E$$ and $$C F$$ are drawn to opposite sides $$B C , C A$$ and $$A B$$ respectively which meet at a point $$O.$$ Now triangle $$DEF$$ is completed, also length $$OA, OB$$ and $$OC$$ respectively are $$p , q$$ and $$r.$$
Radius of the circumcircle of $$\Delta DEF$$ is

A
$$R/2$$

B
$$R$$

C
$$R/4$$

D
None of these

Solution

$$\begin{array}{l} Let\, \, circumradius\, \, be\, \, 'R' \\ 2R'=\frac { { EF } }{ { \sin \angle E-D } } \\ =\frac { { a\cos A } }{ { \sin \left( { \pi -2A } \right) } } \\ =\frac { a }{ { 2\sin A } } \\ =R \\ R'=\frac { R }{ 2 } \\ Hence,\, option\, A\, is\, the\, correct\, answer. \end{array}$$
Question 9
1 / -0

Let $$C_1$$ and $$C_2$$ are circles defined by $$x^2+y^2-20x+64=0$$ and $$x^2+y^2+30x+144=0$$. The length of the shortest line segment PQ that is tangent to $$C_1$$ at P and to $$C_2$$ at Q is?

A
$$15$$

B
$$18$$

C
$$20$$

D
$$24$$

Solution
Question 10
1 / -0

The value of c for which $$x+y=2$$ touches circle $$x^2+y^2-2cx=0$$ and $$(1, 0)$$ lies inside the circle, is?

A
$$2$$

B
$$\sqrt{2}-1$$

C
$$2(\sqrt{2}-1)$$

D
$$2(\sqrt{2}+1)$$

Solution