Equation of Tangent and Normal

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JEE Main 2010

The circle $x^{2} + y^{2} = 4 x + 8 y + 5$ intersects the line $3 x - 4 y = m$ at two distinct points if

- 35 < m < 15

(B)

15 < m < 65

(C)

35 < m < 85

(D)

- 85 < m < - 35

JEE Main 2002

LEVELJEE Advanced

If the chord $y = m x + 1$ of the circle $x^{2} + y^{2} = 1$ subtends an angle of measure $4 5^{\circ}$ at the major segment of the circle then value of $m$ is

(A)

2 \pm 2

(B)

- 2 \pm 2

(C)

- 1 \pm 2

The circle $C_{1} : x^{2} + y^{2} = 3$ , with centre at $O$ , intersects the parabola $x^{2} = 2 y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_{1}$ at $P$ touches other two circles $C_{2}$ and $C_{3}$ at $R_{2}$ and $R_{3}$ , respectively. Suppose $C_{2}$ and $C_{3}$ have equal radii $23$ and centres $Q_{2}$ and $Q_{3}$ , respectively. If $Q_{2}$ and $Q_{3}$ lie on the $y$ -axis, then

* Multiple Correct Options

(A)

Q_{2} Q_{3} = 12

(B)

R_{2} R_{3} = 46

(C)

area of the triangle

Δ O R_{2} R_{3}

62

(D)

area of the triangle

Δ P Q_{2} Q_{3}

42

JEE Advanced 2016

LEVELJEE Advanced

Let $R S$ be the diameter of the circle $x^{2} + y^{2} = 1$ , where $S$ is the point $(1, 0)$ . Let $P$ be a variable point (other than $R$ and $S$ ) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$ . The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $R S$ at point $E$ . Then the locus of $E$ passes through the point(s)

* Multiple Correct Options

(A)

(\frac{1}{3}, \frac{1}{3})

(B)

(\frac{1}{4}, \frac{1}{2})

(C)

(\frac{1}{3}, - \frac{1}{3})

(D)

(\frac{1}{4}, - \frac{1}{2})

JEE Advanced 2009

LEVELJEE Main

Tangents drawn from the point $P (1, 8)$ to the circle $x^{2} + y^{2} - 6 x - 4 y - 11 = 0$ touch the circle at the points $A$ and $B$ . The equation of the circumcircle of the triangle $P A B$ is

(A)

x^{2} + y^{2} + 4 x - 6 y + 19 = 0

(B)

x^{2} + y^{2} - 4 x - 10 y + 19 = 0

(C)

x^{2} + y^{2} - 2 x + 6 y - 29 = 0

(D)

x^{2} + y^{2} - 6 x - 4 y + 19 = 0

JEE Advanced 2007

LEVELJEE Advanced

Let $A B C D$ be a quadrilateral with area 18, with side $A B$ parallel to the side $C D$ and $2 A B = C D$ . Let $A D$ be perpendicular to $A B$ and $C D$ . If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is

(A)

(B)

(C)

3/2

(D)

JEE Advanced 2005

LEVELJEE Main

Tangent to the curve $y = x^{2} + 6$ at a point $(1, 7)$ touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ at a point $Q$ . Then the coordinates of $Q$ are

(A)

(- 6, - 11)

(B)

(- 9, - 13)

(C)

(- 10, - 15)

(D)

(- 6, - 7)

JEE Advanced 2002

LEVELJEE Main

If $a > 2 b > 0$ then the positive value of $m$ for which $y = m x - b 1 + m^{2}$ is a common tangent to $x^{2} + y^{2} = b^{2}$ and $(x - a)^{2} + y^{2} = b^{2}$ is

(A)

\frac{2 b}{a ^{2} - 4 b ^{2}}

(B)

\frac{a ^{2} - 4 b ^{2}}{2 b}

(C)

\frac{2 b}{a - 2 b}

(D)

\frac{b}{a - 2 b}

JEE Advanced 1998

LEVELJEE Main

The number of common tangents to the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} - 6 x - 8 y = 24$ is

The equations of the tangents drawn from the origin to the circle $x^{2} + y^{2} - 2 r x - 2 h y + h^{2} = 0$ , are

* Multiple Correct Options

(A)

x = 0

(B)

y = 0

(C)

(h^{2} - r^{2}) x - 2 r h y = 0

(D)

(h^{2} - r^{2}) x + 2 r h y = 0

LEVELJEE Advanced

Let $P Q$ and $R S$ be tangents at the extremities of the diameter $P R$ of a circle of radius $r$ . If $P S$ and $R Q$ intersect at a point $X$ on the circumference of the circle, then $2 r$ equals

(A)

P Q \cdot R S

(B)

(P Q + R S) /2

(C)

2 P Q \cdot R S / (P Q + R S)

(D)

(P Q^{2} + R S^{2}) /2

LEVELJEE Main

The angle between a pair of tangents drawn from a point $P$ to the circle $x^{2} + y^{2} + 4 x - 6 y + 9 sin^{2} α + 13 cos^{2} α = 0$ is $2 α$ . The equation of the locus of the point $P$ is

(A)

x^{2} + y^{2} + 4 x - 6 y + 4 = 0

(B)

x^{2} + y^{2} + 4 x - 6 y - 9 = 0

(C)

x^{2} + y^{2} + 4 x - 6 y - 4 = 0

(D)

x^{2} + y^{2} + 4 x - 6 y + 9 = 0

LEVELJEE Main

The centre of the circle passing through the point $(0, 1)$ and touching the curve $y = x^{2}$ at $(2, 4)$ is

(A)

(\frac{- 16}{5}, \frac{27}{10})

(B)

(\frac{- 16}{7}, \frac{53}{10})

(C)

(\frac{- 16}{5}, \frac{53}{10})

The centres of two circles $C_{1}$ and $C_{2}$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segement joining the centres of $C_{1}$ and $C_{2}$ and $C$ be a circle touching circles $C_{1}$ and $C_{2}$ externally. If a common tangent to $C_{1}$ and $C$ passing through $P$ is also a common tangent to $C_{2}$ and $C$ , then the radius of the circle $C$ is

JEE Advanced 2005

LEVELJEE Advanced

Circles with radii 3, 4 and 5 touch each other externally. If $P$ is the point of intersection of tangents to these circles at their points of contact, find the distance of $P$ from the points of contact.

JEE Advanced 2001

LEVELJEE Advanced

Let $2 x^{2} + y^{2} - 3 x y = 0$ be the equation of a pair of tangents drawn from the origin $O$ to a circle of radius 3 with centre in the first quadrant. If $A$ is one of the points of contact, find the length of $O A$ .

JEE Advanced 1984

LEVELJEE Main

The lines $3 x - 4 y + 4 = 0$ and $6 x - 8 y - 7 = 0$ are tangents to the same circle. The radius of this circle is .........

JEE Advanced 1981

LEVELJEE Main

Let $A$ be the centre of the circle $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ . Suppose that the tangents at the points $B (1, 7)$ and $D (4, - 2)$ on the circle meet at the point $C$ . Find the area of the quadrilateral $A B C D$ .

JEE Advanced 1999

LEVELJEE Advanced

Let $T_{1}, T_{2}$ be two tangents drawn from $(- 2, 0)$ onto the circle $C : x^{2} + y^{2} = 1$ . Determine the circles touching $C$ and $T_{1}, T_{2}$ as their pair of tangents. Further, find the equations of all possible common tangents to these circles, when taken two at a time.

JEE Advanced 1996

LEVELJEE Advanced

Find the intervals of values of $a$ for which the line $y + x = 0$ bisects two chords drawn from a point $(\frac{1 + 2 a}{2}, \frac{1 - 2 a}{2})$ to the circle $2 x^{2} + 2 y^{2} - (1 + 2 a) x - (1 - 2 a) y = 0$ .

JEE Advanced 1993

LEVELJEE Advanced

Find the coordinates of the point at which the circles $x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ and $x^{2} + y^{2} - 12 x - 8 y - 36 = 0$ touch each other. Also find equations common tangents touching the circles in the distinct points.

JEE Advanced 1992

LEVELJEE Advanced

Three circles touch one another externally. The tangent at their point of contact meet at a point whose distance from a point of contact is $4$ . Find the ratio of the product of the radii to the sum of the radii of the circles.

JEE Advanced 1992

LEVELJEE Main

Let a circle be given by $2 x (x - a) + y (2 y - b) = 0, (a \neq = 0, b \neq = 0)$ . Find the condition on $a$ and $b$ if two chords, each bisected by the $x$ -axis, can be drawn to the circle from $(a, b /2)$ .

JEE Advanced 1991

LEVELJEE Main

Two circles, each of radius 5 units, touch each other at $(1, 2)$ . If the equation of their common tangent is $4 x + 3 y = 10$ , find the equation of the circles.

JEE Advanced 1990

LEVELJEE Advanced

A circle touches the line $y = x$ at a point $P$ such that $O P = 42$ , where $O$ is the origin. The circle contains the point $(- 10, 2)$ in its interior and the length of its chord on the line $x + y = 0$ is $62$ . Determine the equation of the circle.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The circle x2+y2=4x+8y+5 intersects the line 3x−4y=m at two distinct points if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the chord y=mx+1 of the circle x2+y2=1 subtends an angle of measure 45∘ at the major segment of the circle then value of m is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tangents drawn from the point P(1,8) to the circle x2+y2−6x−4y−11=0 touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tangent to the curve y=x2+6 at a point (1,7) touches the circle x2+y2+16x+12y+c=0 at a point Q. Then the coordinates of Q are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a>2b>0 then the positive value of m for which y=mx−b1+m2​ is a common tangent to x2+y2=b2 and (x−a)2+y2=b2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of common tangents to the circles x2+y2=4 and x2+y2−6x−8y=24 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equations of the tangents drawn from the origin to the circle x2+y2−2rx−2hy+h2=0, are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The angle between a pair of tangents drawn from a point P to the circle x2+y2+4x−6y+9sin2α+13cos2α=0 is 2α. The equation of the locus of the point P is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The centre of the circle passing through the point (0,1) and touching the curve y=x2 at (2,4) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The lines 3x−4y+4=0 and 6x−8y−7=0 are tangents to the same circle. The radius of this circle is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two circles, each of radius 5 units, touch each other at (1,2). If the equation of their common tangent is 4x+3y=10, find the equation of the circles.

The circle $x^{2} + y^{2} = 4 x + 8 y + 5$ intersects the line $3 x - 4 y = m$ at two distinct points if

If the chord $y = m x + 1$ of the circle $x^{2} + y^{2} = 1$ subtends an angle of measure $4 5^{\circ}$ at the major segment of the circle then value of $m$ is

Tangents drawn from the point $P (1, 8)$ to the circle $x^{2} + y^{2} - 6 x - 4 y - 11 = 0$ touch the circle at the points $A$ and $B$ . The equation of the circumcircle of the triangle $P A B$ is

Tangent to the curve $y = x^{2} + 6$ at a point $(1, 7)$ touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ at a point $Q$ . Then the coordinates of $Q$ are

If $a > 2 b > 0$ then the positive value of $m$ for which $y = m x - b 1 + m^{2}$ is a common tangent to $x^{2} + y^{2} = b^{2}$ and $(x - a)^{2} + y^{2} = b^{2}$ is

The number of common tangents to the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} - 6 x - 8 y = 24$ is

The equations of the tangents drawn from the origin to the circle $x^{2} + y^{2} - 2 r x - 2 h y + h^{2} = 0$ , are

The angle between a pair of tangents drawn from a point $P$ to the circle $x^{2} + y^{2} + 4 x - 6 y + 9 sin^{2} α + 13 cos^{2} α = 0$ is $2 α$ . The equation of the locus of the point $P$ is

The centre of the circle passing through the point $(0, 1)$ and touching the curve $y = x^{2}$ at $(2, 4)$ is

The lines $3 x - 4 y + 4 = 0$ and $6 x - 8 y - 7 = 0$ are tangents to the same circle. The radius of this circle is .........

Two circles, each of radius 5 units, touch each other at $(1, 2)$ . If the equation of their common tangent is $4 x + 3 y = 10$ , find the equation of the circles.