# Position of a Point with Respect to Circle

## Trending Questions

**Q.**Let T be the line passing through the points P(−2, 7) and Q(2, −5). Let F1 be the set of all pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point, say, M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1. Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point R(1, 1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then, which of the following statement(s) is (are) TRUE?

- The point (−2, 7) lies in E1.
- The point (45, 75) does NOT lie in E2.
- The point (12, 1) lies in E2.
- The point (0, 32) does NOT lie in E1.

**Q.**The range of parameter a for which the variable line y=2x+a lies between the circles x2+y2−2x−2y+1=0 and x2+y2−16x−2y+61=0 without intersecting or touching either circle is

- (−15, −1)
- (√5−1, ∞)
- (−∞, −2√5−15)
- (2√5−15, −√5−1)

**Q.**

If OA and OB be the tangents to the circle x2+y2−6x−8y+21=0 drawn from the origin O, then AB =

11

12

**Q.**x−x1cosθ=y−y1sinθ=r, represents:

- Equation of straight line, if θ is variable & r is constant
- Equation of circle, if r is variable & θ is constant
- Equation of circle, if θ is variable & r is constant
- Equation of straight line, if r is variable & θ is constant

**Q.**The nearest point on the circle x2+y2−6x+4y−12=0 from the point P(−5, 4) is Q(α, β), then the value of α+β is

- 0
- 4
- 2
- 7

**Q.**The range of values of α for which the point (α, α) lies in the interior part of smaller segment of x2+y2=4 intercepted by the line 3x+4y+7=0, is

- (−∞, √2)
- (−√2, √2)
- (−√2, −1)
- ϕ

**Q.**The range of values of r, for which the point (−5+r√2, −3+r√2) is an interior point of the major segment of the circle x2+y2=16 cut-off by the line x+y=2, is

- (4√2−√14, 5√2)
- (4√2−√14, 4√2+√14)
- None of the above
- (−∞, 5√2)

**Q.**The number of tangents which can be drawn from the point (–1, 2) to the circle x2+y2+2x−4y+4=0 is

- 1
- 2
- 0
- 3

**Q.**If P(0, 0), Q(1, 0) and R(12, √32) are three given points, then the centre of the circle for which the lines PQ, QR and RP are the tangents is

- (12, 14)
- (12, 12√3)
- (12, √34)
- (12, −1√3)

**Q.**If a line makes an angles α, β, γ with positive axes. Then the range of sinαsinβ+sinβsinγ+sinαsinγ is:

- [−12, 1]
- [12, 2]
- [−1, 2]
- (−1, 2]

**Q.**If d1 and d2 are the longest and the shortest distances of the point P(−7, 2) from the circle x2+y2−10x−14y−51=0, then the value of d21+d22 is

**Q.**

The minimum radius vector of the curve $\frac{{a}^{2}}{{x}^{2}}+\frac{{b}^{2}}{{y}^{2}}=1$ is of length

$a-b$

$\left|a+b\right|$

$2a+b$

None of these

**Q.**If number of integral coordinates (x, y) which lie inside to the circle x2+y2=25 is n, then [n9] is

( [.] represents the greatest integer function. )

**Q.**The set of values of a for which the point(a−1, a+1) lies outside the circle x2+y2=8 and inside the circle x2+y2−12x+12y−62=0 is

- (−3√2, −√3)∪(√3, 3√2)
- (−2√2, −√3)∪(√3, 3√2)
- (−3√2, √3)∪(√3, 3√2)
- (−3√2, −√3)∪(−√3, 3√2)

**Q.**The area of the loop of the curve y2=x4(x+2) is [in square units]

- 32√2105
- 64√2105
- 128√2105
- 256√2105

**Q.**If (a, 0) is an endpoint of a diameter of the circle x2+y2=4, then x2−4x−a2=0 has

- exactly one real root in (−1, 0]
- exactly one real root in [2, 5]
- distinct roots greater than −1
- distinct roots less than 5

**Q.**A tower subtends angles θ, 2θ and 3θ at three points A, B, C respectively lying on a horizontal line through the foot of tower. Then the ratio ABBC equals

- sin3θsinθ
- sinθsin3θ
- cos3θcosθ
- tanθtan3θ

**Q.**From the point P(2, 1), a line of slope m∈R is drawn so as to cut the circle x2+y2=1 in points A and B. If the slope m is varied, then the greatest possible value of PA+PB is

- 2√5
- 2√5
- 1√5
- 10√5

**Q.**Let T be the line passing through the points P(−2, 7) and Q(2, −5). Let F1 be the set of all pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point, say, M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1. Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point R(1, 1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then, which of the following statement(s) is (are) TRUE?

- The point (−2, 7) lies in E1.
- The point (45, 75) does NOT lie in E2.
- The point (12, 1) lies in E2.
- The point (0, 32) does NOT lie in E1.

**Q.**For a>0, let the curves C1:y2=ax and C2:x2=ay intersect at origin O and a point P. Let the line x=b (0<b<a) intersects the chord OP and the x-axis at points Q and R, respectively. If the line x=b bisects the area bounded by the curves, C1 and C2, and the area of ΔOQR=12, then 'a' satisfies the equation:

- x6−12x3+4=0
- x6−12x3−4=0
- x6+6x3−4=0
- x6−6x3+4=0

**Q.**The equation of the image of the circle x2+y2−6x−4y=0 in the bisector of 2nd and 4th quadrant is

- x2+y2+4x−6y=0
- x2+y2−4x+6y=0
- x2+y2+4x+6y=0
- x2+y2−4x−6y=0

**Q.**Let C:x2+y2−x−y−6=0 and point P(a−1, a+1) lies inside the circle C. If the line x+y−2=0 divides the circle in two segments, then

- P to lie in the larger segment of the intersection if a∈(−1, 1)
- P to lie in the larger segment of the intersection if a∈(1, 2)
- P to lie in the smaller segment of the intersection if a∈(1, 2)
- P to lie in the smaller segment of the intersection if a∈(−1, 1)

**Q.**The greatest distance of the point P(9, 7), from the circle x2+y2−2x−2y−23=0, is

**Q.**The orthocentre of the triangle formed by the lines x+y=1, 2x+3y=6 and 4x−y+4=0 lies in

- first quadrant
- second quadrant
- third quadrant
- fourth quadrant

**Q.**A point equidistant from the line 4x + 3y +10 = 0, 5x - 12y + 26 = 0 and 7x + 24y - 50 = 0 is ?

**Q.**For the circle x2+y2+6x+8y=0, and the points P(−3, −6) and Q(4, −2),

- Q lies inside and P lies outside the circle
- P lies inside and Q lies outside the circle
- P, Q lie outside the circle
- P, Q lie inside the circle

**Q.**→p=w^i+x^j and →q=y^i+z^j are two vectors in the first quadrant such that |→p|=2|→q|=2r, r>0 and →p⋅→q=0. If →a=w^i+2y^j and →b=x2^i+z^j, then

- |→a|=r
- |→b|=r
- →a⋅→b=0
- |→a×→b|=2r

**Q.**From the point P(2, 1), a line of slope m∈R is drawn so as to cut the circle x2+y2=1 in points A and B. If the slope m is varied, then the greatest possible value of PA+PB is

- 2√5
- 10√5
- 2√5
- 1√5

**Q.**How many words can be formed from the letters of the word TRIANGLE ? In how many of these does the word start with T and end with E ?

**Q.**If d1 and d2 are the longest and the shortest distances of the point P(−7, 2) from the circle x2+y2−10x−14y−51=0, then the value of d21+d22 is