Mathematics - Circle
Exam Duration: 45 Mins Total Questions : 30
The equation of the circle which passes through the origin and makes intercept of lengths a and b on the X and Y-axes respectively, are
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }\pm ax\pm by=0\)
- (b)
\({ x }^{ 2 }\pm { y }^{ 2 }=0\)
- (c)
\({ x }^{ 2 }\pm ax\pm by=0\)
- (d)
None of the above
The circle \({ x }^{ 2 }+{ y }^{ 2 }={ r }^{ 2 }\) intersects the line Y = mX + c at the two real distinct points, if
- (a)
\(-r\sqrt { 1+{ m }^{ 2 } }\)
- (b)
\(-c\sqrt { 1-{ m }^{ 2 } }\)
- (c)
\(-r\sqrt { 1-{ m }^{ 2 } }\)
- (d)
None of the above
Let \({ X }^{ 2 }+{ Y }^{ 2 }+2gX+2fY+c=0\) be an equation of circle.
Column I | Column II |
A. If circle lies in first quadrant, then | p. g < 0 |
B. If circle lies above X-axis, then | q. g > 0 |
C. If circle lies on the left of Y-axis, then | r. g2 - c < 0 |
D. If circle touches positive X-axis and does not intersect Y.axis then | s. c > 0 |
- (a)
A B C D prs rs qs ps - (b)
A B C D pqr sq pr ps - (c)
A B C D pr ps pq qrs - (d)
None of the above
X - 2Y + 3 = 0 and 4X - 3Y + 2 = 0 are two diameters of a circle. If radius is equal to 1, then equation of the circle is
- (a)
\({ \left( x-1 \right) }^{ 2 }+{ \left( y-2 \right) }^{ 2 }=1\)
- (b)
\({ \left( x-2 \right) }^{ 2 }+{ \left( y-1\right) }^{ 2 }=1\)
- (c)
\({ x }^{ 2 }+{ y }^{ 2 }+2x+4y+4=0\)
- (d)
\({ x }^{ 2 }+{ y }^{ 2 }-3x-4y+7=0\)
What is the length of an equilateral triangle inscribed in the circle \({ x }^{ 2 }+{ y }^{ 2 }=\frac { 4 }{ 3 } ?\)
- (a)
2 units
- (b)
5 units
- (c)
3 units
- (d)
7 units
The range of values of m for which the line Y = mX + 2 cuts the circle \({ X }^{ 2 }+{ Y }^{ 2 }=1\) at distinct or coincident point is
- (a)
\(\left( -\infty ,-\sqrt { 3] } \cup [\sqrt { 3 } ,\infty \right) \)
- (b)
\(\left[ -\sqrt { 3 } ,\sqrt { 3 } \right] \)
- (c)
\([\sqrt { 3 } ,\infty )\)
- (d)
None of the above
The centre of the circle inscribed in square formed by the lines \({ X }^{ 2 }-8X+12=0\) and \({ Y }^{ 2 }-14Y+45=0\) is
- (a)
(4, 7)
- (b)
(7, 4)
- (c)
(9, 4)
- (d)
(4, 9)
Line lX + mY + 1 = 0, touches a fixed circle. Also, \({ 4l }^{ 2 }-{ 5m }^{ 2 }+6l+1=0\), then
- (a)
the centre of the circle is at the point (4, 0)
- (b)
the radius of the circle is equal to \(\sqrt{5}\)
- (c)
the circle passes through origin
- (d)
None of the above
If the lines 3X - 4Y - 7 = 0 and 2X - 3Y - 5 = 0 are two diameters of a circle of area 49\(\pi\) sq units, then equation of the circle is
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }+2x-2y-62=0\)
- (b)
\({ x }^{ 2 }+{ y }^{ 2 }-2x+2y-62=0\)
- (c)
\({ x }^{ 2 }+{ y }^{ 2 }-2x+2y-47=0\)
- (d)
\({ x }^{ 2 }+{ y }^{ 2 }+2x-2y-47=0\)
If a circle having centre at \(\left( \alpha ,\beta \right) \) radius r completely lies with in two lines x + y = 2 and x + y = - 2 , then min \(\left( \left| \alpha +\beta +2 \right| ,\left| \alpha +\beta -2 \right| \right) \) is
- (a)
greater than \(\sqrt { 2 } r\)
- (b)
less than \(\sqrt { 2 } r\)
- (c)
greater than 2r
- (d)
less than 2r
If the distances from the origin of the centres of three circles \({ x }^{ 2 }+{ y }^{ 2 }+2{ \lambda }_{ i }x-{ c }^{ 2 }\) = 0 (i = 1, 2, 3) are in GP, then the lengths of the tangents drawn to them from any point on the circle x2 + y2 = c2 are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
none of these
The equation of a tangent to the circle x2 + y2 = 25 passing through (-2, 11) is
- (a)
4x + 3y = 25
- (b)
3x + 4y = 38
- (c)
24x-7y+125=0
- (d)
7x+24y=230
The circle x2+y2+4x- 7y + 12 = 0 cuts an intercept on y-axis of length
- (a)
3
- (b)
4
- (c)
7
- (d)
1
The number of integral values of λ for which x2+y2+入x+(1-入)y+5=0 is the equation of a circle whose radius cannot exceed 5, is
- (a)
14
- (b)
18
- (c)
16
- (d)
none of these
If the line 3x - 4y -⋋=0 touches the circle x2+y2-4x - 8y - 5 = 0 at (a, b), then ⋋+a+b is equal to
- (a)
20
- (b)
22
- (c)
-30
- (d)
-28
If 7l2 - 9m2 + 81 + 1 = 0 and we have to find equation of circle having Ix + my + 1 = 0 is a tangent and we can adjust given condition as 16l2 + 8l + 1 = 9 (l2 + m2)
or (4l+1)2=9(l2+m2)⇒ \(\frac { \left| 4l+1 \right| }{ \sqrt { \left( { l }^{ 2 }+{ m }^{ 2 } \right) } } =3\)
Centre of circle = (4, 0) and radius = 3 when any two non parallel lines touching a circle, then centre of circle lies on angle bisector of lines.
If 16m2-8l-1=0 then equation of the circle having Ix + my + 1=0 is a tangent is
- (a)
x2+y2+8x=0
- (b)
x2+y2-8x=0
- (c)
x2+y2+8y=0
- (d)
x2+y2-8y=0