Q.

A variable straight line of slope 4 intersects the hyperbola xy = 1 at two points. The locus of the point which divides the line segment between these two points in the ratio 1 : 2 is

Q.

With respect to position of two circles which of the following statements is/are correct?

1. If one circle lies completely outside the other circle,

Number of direct common tangents = 2

Number of transverse common tangents = 2

2. If two circles touch each other externally

Number of direct common tangents = 2

Number of transverse common tangents = 1

3. If two circles touch each other internally

Number of direct common tangents = 1

Number of transverse common tangents = 0

4. If two circles intersect each other at two points

Number of direct common tangents = 2

Number of transverse common tangents = 0

5. If one circle lies completely inside the other circle

Number of direct common tangents = 0

Number of transverse common tangents = 0

Only 1, 2, 3
Only 1, 2, 5
Only 1, 2, 3, 4
All 1, 2, 3, 4, 5

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Q. If

C_{1}

and

C_{2}

are circles whose equations are

x^{2} + y^{2} - 20 x + 64 = 0

and

x^{2} + y^{2} + 30 x + 144 = 0

, then the length of the shortest line segment

P Q

that is tangent to

C_{1}

at

P

and to

C_{2}

at

Q

is

$15 units$
$18 units$
$20 units$
$24 units$

View Solution

Q. For circles

(x - 3)^{2} + (y - 2)^{2} = 9

and

(x + 5)^{2} + (y + 6)^{2} = 9

which among the following are tangent equations

$y + 2 = \frac{16 - 3 \sqrt{23}}{7} (x + 1)$
$y + 2 = \frac{16 + 3 \sqrt{23}}{7} (x + 1)$
$y + 1 = x + 3 \sqrt{2}$
$y + 1 = x - 3 \sqrt{2}$

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Q.

Let $c_{1} & c_{2}$ be the centers and $r_{1} & r_{2}$ be the radius of two circles. Then

Cases Conditions

p. 1. $| r_{1} - r_{2} | < c_{1} c_{2} < r_{1} + r_{2}$

q. 2. $| r_{1} - r_{2} | = c_{1} c_{2}$

r. 3. $c_{1} c_{2} < | r_{1} + r_{2} |$

s.

4. $r_{1} + r_{2} < c_{1} r_{2}$

t. 5. $r_{1} + r_{2} = c_{1} r_{2}$

p - 5, q - 4 , r - 1, s - 2 , t - 3
p - 4, q - 5 , r - 1, s - 3 , t - 2
p - 1, q - 2 , r - 3, s - 4 , t - 5
p - 4, q - 5 , r - 1 , s - 2 , t - 3

View Solution

Q. There are two circles whose equations are

x^{2} + y^{2} = 9

and

x^{2} + y^{2} - 8 x - 6 y + n^{2} = 0, n \in Z

. If the two circles have exactly two common tangents, then the number of possible values of

n

is

$8$
$5$
$2$
$9$

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Q. A line with positive direction cosines passes through the point

P (2, - 1, 2)

and makes equal angles with the coordinate axes. The line meets the plane

2 x + y + z = 9

at point

Q

. The length of the line segment

P Q

equals

1
2
$\sqrt{3}$
$\sqrt{2}$

View Solution

Q. If

a r g \frac{z - 1}{z + 1} = \frac{π}{2}

then the locus of

z

is:

Circle with radius $2$
Circle with radius $1$
Srtight line
Pair of lines

View Solution

Q. The triangle PQR is inscribed in the circle

x^{2} + y^{2} = 25.

If

Q

and

R

have coordinates

(3, 4)

and

(- 4, 3),

respectively, then

∠ Q P R

is equal to

$\frac{π}{2}$
$\frac{π}{3}$
$\frac{π}{4}$
$\frac{π}{6}$

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Q.

Find the Slope of the transverse common tangent to the circles $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$

$\frac{- 5}{12}$
$\frac{5}{12}$
$\frac{12}{5}$
$\frac{- 12}{5}$

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Q.

Find the Slope of the transverse common tangent to the circles $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$

View Solution

Q. For circles

(x - 3)^{2} + (y - 2)^{2} = 9

and

(x + 5)^{2} + (y + 6)^{2} = 9

which among the following are tangent equations

$y + 2 = \frac{16 - 3 \sqrt{23}}{7} (x + 1)$
$y + 2 = \frac{16 + 3 \sqrt{23}}{7} (x + 1)$
$y + 1 = x + 3 \sqrt{2}$
$y + 1 = x - 3 \sqrt{2}$

View Solution

Q.

With respect to position of two circles which of the following statements is/are correct?

1. If one circle lies completely outside the other circle,

Number of direct common tangents = 2

Number of transverse common tangents = 2

2. If two circles touch each other externally

Number of direct common tangents = 2

Number of transverse common tangents = 1

3. If two circles touch each other internally

Number of direct common tangents = 1

Number of transverse common tangents = 0

4. If two circles intersect each other at two points

Number of direct common tangents = 2

Number of transverse common tangents = 0

5. If one circle lies completely inside the other circle

Number of direct common tangents = 0

Number of transverse common tangents = 0

Only 1, 2, 3
Only 1, 2, 5
Only 1, 2, 3, 4
All 1, 2, 3, 4, 5

View Solution

Q.

Let $c_{1} & c_{2}$ be the centers and $r_{1} & r_{2}$ be the radius of two circles. Then

Cases Conditions

p. 1. $| r_{1} - r_{2} | < c_{1} c_{2} < r_{1} + r_{2}$

q. 2. $| r_{1} - r_{2} | = c_{1} c_{2}$

r. 3. $c_{1} c_{2} < | r_{1} + r_{2} |$

s.

4. $r_{1} + r_{2} < c_{1} r_{2}$

t. 5. $r_{1} + r_{2} = c_{1} r_{2}$

p - 4, q - 5 , r - 1, s - 3 , t - 2
p - 5, q - 4 , r - 1, s - 2 , t - 3
p - 1, q - 2 , r - 3, s - 4 , t - 5
p - 4, q - 5 , r - 1 , s - 2 , t - 3

View Solution

Q. Prove:
If two tangents PT1 & PT2 are drawn from the point P(x1, y1) to the circle S = x2 + y2 + 2gx + 2fy + c = 0,
then the equation of the chord of contact T1T2 is: xx1 + yy1 + g (x + x1) + f (y + y1) + c = 0.

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Q.

The equation of the common tangent to the circles $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ at their point of contact.

12x + 5y + 19 = 0
5x + 12y + 19 = 0
5x - 12y + 19 = 0
12x - 5y + 19 = 0

View Solution

Q.

Find the equations of transverse common tangents for two circles

$x^{2} + y^{2} + 6 x - 2 y + 1 = 0, x^{2} + y^{2} - 2 x - 6 y + 9 = 0$

$35 x^{2} + 12 x y - 18 x = 0$
$35 y^{2} + 12 x y - 18 y = 0$
$3 x^{2} - 4 x y + 16 y - 12 x = 0$
$3 y^{2} - 4 x y + 16 x - 12 y = 0$

View Solution

Q. The circles

x^{2} + y^{2} - 2 x - 4 y + 1 = 0

and

x^{2} + y^{2} + 4 x + 4 y - 1 = 0

touch internally
touch externally
have $3 x + 4 y - 1 = 0$ as the common tangent at the point of contact.
have $3 x + 4 y + 1 = 0$ as the common tangent at the point of contact.

View Solution

Q. If tangents be drawn to the circle

x^{2} + y^{2} = 12

at its points of intersection with the circle

x^{2} + y^{2} - 5 x + 3 y - 2 = 0

, find the co-ordinates of their point of intersection.

View Solution

Q. The circles

x^{2} + y^{2} - 2 x - 4 y + 1 = 0

and

x^{2} + y^{2} + 4 x + 4 y - 1 = 0

touch internally
touch externally
have $3 x + 4 y - 1 = 0$ as the common tangent at the point of contact.
have $3 x + 4 y + 1 = 0$ as the common tangent at the point of contact.

View Solution

Q.

The equation of the common tangent to the circles $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ at their point of contact.

$12 x + 5 y + 19 = 0$
$5 x + 12 y + 19 = 0$
$5 x - 12 y + 19 = 0$
$12 x - 5 y + 19 = 0$

View Solution

Q.

Find the equations of transverse common tangents for two circles

$x^{2} + y^{2} + 6 x - 2 y + 1 = 0, x^{2} + y^{2} - 2 x - 6 y + 9 = 0$

35y² + 12xy - 18y = 0
3x² - 4xy + 16y - 12x = 0
3y² - 4xy + 16x - 12y = 0
35x² + 12xy - 18x = 0

View Solution

Q.

Find the equations of transverse common tangents for two circles

$x^{2} + y^{2} + 6 x - 2 y + 1 = 0, x^{2} + y^{2} - 2 x - 6 y + 9 = 0$

$35 x^{2} + 12 x y - 18 x = 0$
$35 y^{2} + 12 x y - 18 y = 0$
$3 x^{2} - 4 x y + 16 y - 12 x = 0$
$3 y^{2} - 4 x y + 16 x - 12 y = 0$

View Solution

Transverse Common Tangent

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