\section{EXERCISE - II (CIRCLE)}

\section{SINGLE CORRECT TYPE}

1. If \(a\) be the radius of a circle which touches \(x\)-axis at the origin, then its equation is

(A) \(x^{2}+y^{2}+a x=0\)

(C) \(x^{2}+y^{2} \pm 2 x a=0\)

(B) \(x^{2}+y^{2} \pm 2 y a=0\)

(D) \(x^{2}+y^{2}+y a=0\)

2. The equation of the circle which touches the axis of \(y\) at the origin and passes through \((3,4)\) is

(A) \(4\left(x^{2}+y^{2}\right)-25 x=0\)

(B) \(3\left(x^{2}+y^{2}\right)-25 x=0\)

(C) \(2\left(x^{2}+y^{2}\right)-3 x=0\)

(D) \(4\left(x^{2}+y^{2}\right)-25 x+10=0\)

3. The equation to the circle whose radius is 4 and which touches the negative \(\mathrm{x}\)-axis at a distance 3 units from the origin is

(A) \(x^{2}+y^{2}-6 x+8 y-9=0\)

(B) \(x^{2}+y^{2} \pm 6 x-8 y+9=0\)

(C) \(x^{2}+y^{2}+6 x \pm 8 y+9=0\)

(D) \(x^{2}+y^{2} \pm 6 x-8 y-9=0\)

4. \(y=\sqrt{3} x+c_{1} \& y=\sqrt{3} x+c_{2}\) are two parallel tangents of a circle of radius 2 units, then \(\left|c_{1}-c_{2}\right|\) is equal to

(A) 8

(B) 4

(C) 2

(D) 1 9. The gradient of the tangent line at the point \((a \cos \alpha, a \sin \alpha)\) to the circle \(x^{2}+y^{2}=a^{2}\), is

(A) \(\tan (\pi-\alpha)\)

(B) \(\tan \alpha\)

(C) \(\cot \alpha\)

(D) \(-\cot \alpha\)

10. If \(y=c\) is a tangent to the circle \(x^{2}+y^{2}-2 x+2 y-2=0\) at \((1,1)\), then the value of \(c\) is

(A) 1

(B) 2

(C) \(-1\)

(D) \(-2\)

Sol.

12. The greatest distance of the point \(P(10,7)\) from the circle \(x^{2}+y^{2}-4 x-2 y-20=0\) is

(A) 5

(B) 15

(C) 10

(D) none of these

Sol.

Sol.

13. The length of the tangent drawn from the point \((2,3)\) to the circles \(2\left(x^{2}+y^{2}\right)-7 x+9 y-11=0\).

(A) 18

(B) 14

(C) \(\sqrt{14}\)

(D) \(\sqrt{28}\)

Sol.

11. Line \(3 x+4 y=25\) touches the circle \(x^{2}+y^{2}=25\) at the point

(A) \((4,3)\)

(B) \((3,4)\)

(C) \((-3,-4)\)

(D) none of these 18. The locus of the centre of a circle which touches externally the circle, \(x^{2}+y^{2}-6 x-6 y+14=0\) and also touches the \(y\)-axis is given by the equation

(A) \(x^{2}-6 x-10 y+14=0\)

(C) \(y^{2}-6 x-10 y+14=0\)

(B) \(x^{2}-10 x-6 y+14=0\)

(D) \(y^{2}-10 x-6 y+14=0\)

Sol. Sol.

20. A circle is drawn touching the \(x\)-axis and centre at the point which is the reflection of \((a, b)\) in the line \(y-x=0\). The equation of the circle is

(A) \(x^{2}+y^{2}-2 b x-2 a y+a^{2}=0\)

(B) \(x^{2}+y^{2}-2 b x-2 a y+b^{2}=0\)

(C) \(x^{2}+y^{2}-2 a x-2 b y+b^{2}=0\)

(D) \(x^{2}+y^{2}-2 a x-2 b y+a^{2}=0\)

Sol. 19. The common chord of two intersecting circles \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) can be seen from their centres at the angles of \(90^{\circ}\) and \(60^{\circ}\) respectively. If the distance between their centres is equal to \(\sqrt{3}+1\) then the radius of \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) are

(A) \(\sqrt{3}\) and 3

(B) \(\sqrt{2}\) and \(2 \sqrt{2}\)

(C) \(\sqrt{2}\) and 2

(D) \(2 \sqrt{2}\) and 4 25. What is the length of shortest path by which one can go from \((-2,0)\) to \((2,0)\) without entering the interior of circle, \(x^{2}+y^{2}=1\)

(A) \(2 \sqrt{3}\)

(B) \(\sqrt{3}+\frac{2 \pi}{3}\)

(C) \(2 \sqrt{3}+\frac{\pi}{3}\)

(D) none of these

Sol.

26. Three equal circles each of radius \(r\) touch one another. The radius of the circle touching all the three given circle internally is

(A) \((2+\sqrt{3}) r\)

(B) \(\frac{(2+\sqrt{3})}{\sqrt{3}} r\)

(C) \(\frac{(2-\sqrt{3})}{\sqrt{3}} r\)

(D) \((2-\sqrt{3}) r\)

Sol. 27. In a right triangle \(A B C\), right angled at \(A\), on the leg \(A C\) as diameter, a semicircle is described. The chord joining \(A\) with the point of intersection \(D\) of the hypotenuse and the semicircle, then the length \(A C\) equals to

(A) \(\frac{A B \cdot A D}{\sqrt{A B^{2}+A D^{2}}}\)

(B) \(\frac{A B \cdot A D}{A B+A D}\)

(C) \(\sqrt{A B \cdot A D}\)

(D) \(\frac{A B \cdot A D}{\sqrt{A B^{2}-A D^{2}}}\)

Sol.

28. The locus of the centers of the circles which cut the circles \(x^{2}+y^{2}+4 x-6 y+9=0\) and \(x^{2}+y^{2}-5 x+4 y-2=0\) orthogonally is

(A) \(9 x+10 y-7=0\)

(C) \(9 x-10 y+11=0\)

(B) \(x-y+2=0\)

(D) \(9 x+10 y+7=0\) Sol.

\section{MULTIPLE CORRECT TYPE}

33. For the circles \(S_{1} \equiv x^{2}+y^{2}-4 x-6 y-12=0\) and \(S_{2} \equiv x^{2}+y^{2}+6 x+4 y-12=0\) and the line \(L \equiv x+y=0\) (A) \(L\) is common tangent of \(S_{1}\) and \(S_{2}\)

(B) \(L\) is common chord of \(S_{1}\) and \(S_{2}\)

(C) \(L\) is radical axis of \(S_{1} \& S_{2}\)

(D) \(L\) is Perpendicular to the line joining the centre of \(\mathrm{S}_{1} \& \mathrm{~S}_{2}\)

Sol.

32. Number of points \((x, y)\) having integral coordinates satisfying the condition \(x^{2}+y^{2}<25\) is

(A) 69

(B) 80

(C) 81

(D) 77