Length of Tangent and Chord of Contact

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JEE Advanced 2012

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x - 5 y = 20$ to the circle $x^{2} + y^{2} = 9$ is

20 (x^{2} + y^{2}) - 36 x + 45 y = 0

(B)

20 (x^{2} + y^{2}) + 36 x - 45 y = 0

(C)

36 (x^{2} + y^{2}) - 20 x + 45 y = 0

(D)

36 (x^{2} + y^{2}) + 20 x - 45 y = 0

LEVELJEE Main

If the tangent at the point $P$ on the circle $x^{2} + y^{2} + 6 x + 6 y = 2$ meets a straight line $5 x - 2 y + 6 = 0$ at a point $Q$ on the y-axis, then the length of $P Q$ is

(A)

4

(B)

25

(C)

5

(D)

35

JEE Advanced 2003

LEVELJEE Advanced

For the circle $x^{2} + y^{2} = r^{2}$ , find the value of $r$ for which the area enclosed by the tangents drawn from the point $P (6, 8)$ to the circle and the chord of contact is maximum.

JEE Advanced 1987

LEVELJEE Main

The area of the triangle formed by the tangents from the point $(4, 3)$ to the the circle $x^{2} + y^{2} = 9$ and the line joining their points of contact is .........

JEE Advanced 1985

LEVELJEE Main

Let $x^{2} + y^{2} - 4 x - 2 y - 11 = 0$ be a circle. A pair of tangents from the point $(4, 5)$ with a pair of radii form a quadrilateral of area .........

JEE Advanced 1997

LEVELJEE Main

The chords of contact of the pair of tangents drawn from each point on the line $2 x + y = 4$ to circle $x^{2} + y^{2} = 1$ pass through the point .........

JEE Advanced 1988

LEVELJEE Advanced

If the circle $C_{1} : x^{2} + y^{2} = 16$ intersects another circle $C_{2}$ of radius 5 in such a manner that common chord is of maximum length and has a slope equal to $3/4$ , then the coordinates of the centre of $C_{2}$ are .........

JEE Main 2020 (7 January Shift 2)

LEVELJEE Main

Let the tangents drawn from the origin to the circle, $x^{2} + y^{2} - 8 x - 4 y + 16 = 0$ touch it at the points $A$ and $B$ . The $(A B)^{2}$ is equal to :

(A)

\frac{32}{5}

(B)

\frac{64}{5}

(C)

\frac{52}{5}

(D)

\frac{56}{5}

JEE Main 2020 - 7 Jan (Evening)

LEVELJEE Main

Let the tangents drawn from the origin to the circle, $x^{2} + y^{2} - 8 x - 4 y + 16 = 0$ touch it at the point $A$ and $B$ . The $(A B)^{2}$ is equal to:

(A)

\frac{64}{5}

(B)

\frac{24}{5}

(C)

\frac{8}{5}

(D)

\frac{13}{5}

JEE Main 2020 (3 September Shift 2)

LEVELJEE Main

Let the latus rectum of the parabola $y^{2} = 4 x$ be the common chord to the circles $C_{1}$ and $C_{2}$ each of them having radius $25$ . Then, the distance between the centres of the circles $C_{1}$ and $C_{2}$ is:

(A)

(B)

(C)

85

(D)

45

JEE Main 2021 (31 August Shift 2)

LEVELJEE Main

Let $B$ be the centre of the circle $x^{2} + y^{2} - 2 x + 4 y + 1 = 0$ . Let the tangents at two points $P$ and $Q$ on the circle intersect at the point $A (3, 1)$ . Then $8 \cdot (\frac{area Δ A P Q}{area Δ B P Q})$ is equal to .

JEE Main 2022 (29 June Shift 1)

LEVELJEE Advanced

Let the tangent to the circle $C_{1} : x^{2} + y^{2} = 2$ at the point $M (- 1, 1)$ intersect the circle $C_{2} : (x - 3)^{2} + (y - 2)^{2} = 5$ at two distinct points $A$ and $B$ . If the tangents to $C_{2}$ at the points $A$ and $B$ intersect at $N$ , then the area of the triangle $A N B$ is equal to :

(A)

\frac{1}{2}

(B)

\frac{2}{3}

(C)

\frac{1}{6}

(D)

\frac{5}{3}

JEE Main 2023 (06 April Shift 2)

LEVELJEE Main

If the tangents at the points $P$ and $Q$ on the circle $x^{2} + y^{2} - 2 x + y = 5$ meet at the point $R (\frac{9}{4}, 2)$ , then the area of the triangle $P QR$ is

(A)

\frac{5}{4}

(B)

\frac{13}{8}

(C)

\frac{5}{8}

(D)

\frac{13}{4}

JEE Main 2023 (29 January Shift 2)

LEVELJEE Main

A circle with centre $(2, 3)$ and radius 4 intersects the line $x + y = 3$ at the points $P$ and $Q$ . If the tangents at $P$ and $Q$ intersect at the point $S (α, β)$ , then $4 α - 7 β$ is equal to

JEE Advanced 2019

LEVELJEE Advanced

Let the point $B$ be the reflection of the point $A (2, 3)$ with respect to the line $8 x - 6 y - 23 = 0$ . Let $Γ_{A}$ and $Γ_{B}$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $Γ_{A}$ and $Γ_{B}$ such that both the circles are on the same side of $T$ . If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$ , then the length of the line segment $A C$ is

JEE(ADVANCED)-201

LEVELJEE Main

Comprehension Passage

Let S be the circle in the x-y plane defined by the equation

x^{2} + y^{2} = 4

Question 1:

Let $E_{1} E_{2}$ and $F_{1} F_{2}$ be the chords of $S$ passing through the point $P_{0} (1, 1)$ and parallel to the x-axis and the y-axis, respectively. Let $G_{1} G_{2}$ be the chord of $S$ passing through $P_{0}$ and having slope -1. Let the tangents to $S$ at $E_{1}$ and $E_{2}$ meet at $E_{3}$ , the tangents to $S$ at $F_{1}$ and $F_{2}$ meet at $F_{3}$ , and the tangents to $S$ at $G_{1}$ and $G_{2}$ meet at $G_{3}$ . Then, the points $E_{3}, F_{3}$ , and $G_{3}$ lie on the curve

(A)

x + y = 4

(B)

(x - 4)^{2} + (y - 4)^{2} = 16

(C)

(x - 4) (y - 4) = 4

(D)

x y = 4

Question 2:

Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve

(A)

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

(B)

x^{2/3} + y^{2/3} = 2^{4/3}

(C)

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

(D)

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x−5y=20 to the circle x2+y2=9 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the tangent at the point P on the circle x2+y2+6x+6y=2 meets a straight line 5x−2y+6=0 at a point Q on the y-axis, then the length of PQ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For the circle x2+y2=r2, find the value of r for which the area enclosed by the tangents drawn from the point P(6,8) to the circle and the chord of contact is maximum.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The area of the triangle formed by the tangents from the point (4,3) to the the circle x2+y2=9 and the line joining their points of contact is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let x2+y2−4x−2y−11=0 be a circle. A pair of tangents from the point (4,5) with a pair of radii form a quadrilateral of area .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The chords of contact of the pair of tangents drawn from each point on the line 2x+y=4 to circle x2+y2=1 pass through the point .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the tangents drawn from the origin to the circle, x2+y2−8x−4y+16=0 touch it at the points A and B. The (AB)2 is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the tangents drawn from the origin to the circle, x2+y2−8x−4y+16=0 touch it at the point A and B. The (AB)2 is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the tangents at the points P and Q on the circle x2+y2−2x+y=5 meet at the point R(49​,2), then the area of the triangle PQR is

Comprehension Passage

Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x - 5 y = 20$ to the circle $x^{2} + y^{2} = 9$ is

If the tangent at the point $P$ on the circle $x^{2} + y^{2} + 6 x + 6 y = 2$ meets a straight line $5 x - 2 y + 6 = 0$ at a point $Q$ on the y-axis, then the length of $P Q$ is

For the circle $x^{2} + y^{2} = r^{2}$ , find the value of $r$ for which the area enclosed by the tangents drawn from the point $P (6, 8)$ to the circle and the chord of contact is maximum.

The area of the triangle formed by the tangents from the point $(4, 3)$ to the the circle $x^{2} + y^{2} = 9$ and the line joining their points of contact is .........

Let $x^{2} + y^{2} - 4 x - 2 y - 11 = 0$ be a circle. A pair of tangents from the point $(4, 5)$ with a pair of radii form a quadrilateral of area .........

The chords of contact of the pair of tangents drawn from each point on the line $2 x + y = 4$ to circle $x^{2} + y^{2} = 1$ pass through the point .........

Let the tangents drawn from the origin to the circle, $x^{2} + y^{2} - 8 x - 4 y + 16 = 0$ touch it at the points $A$ and $B$ . The $(A B)^{2}$ is equal to :

Let the tangents drawn from the origin to the circle, $x^{2} + y^{2} - 8 x - 4 y + 16 = 0$ touch it at the point $A$ and $B$ . The $(A B)^{2}$ is equal to:

If the tangents at the points $P$ and $Q$ on the circle $x^{2} + y^{2} - 2 x + y = 5$ meet at the point $R (\frac{9}{4}, 2)$ , then the area of the triangle $P QR$ is