A tangent PT is drawn to the circle x²...

MathematicsEquation of Tangent and NormalJEE Advanced 2012Moderate

Comprehension Passage

A tangent

P T

is drawn to the circle

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

at the point

P (3, 1)

. A straight line

L

, perpendicular to

P T

is a tangent to the circle

(x - 3)^{2} + y^{2} = 1

Question 1:

A possible equation of $L$ is

(A)

x - 3 y = 1

(B)

x + 3 y = 1

(C)

x - 3 y = - 1

(D)

x + 3 y = 5

Answer:A

Question 2:

A common tangent of the two circles is

(A)

x = 4

(B)

y = 2

(C)

x + 3 y = 4

(D)

x + 22 y = 6

Answer:D

Visualized Solution (English)

Equation of Tangent .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P T

Circle $C_{1} : x^{2} + y^{2} = 4$ with center $O (0, 0)$ and radius $r_{1} = 2$ .
Point $P$ is $(3, 1)$ .
Equation of tangent at $(x_{1}, y_{1})$ is $x x_{1} + y y_{1} = a^{2}$ .
Substituting $P$ : $x (3) + y (1) = 4 \Rightarrow 3 x + y = 4$ .

Defining Line .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L

Slope of $P T$ ( $m_{P T}$ ) is $- 3$ .
Line $L$ is perpendicular to $P T$ , so its slope $m_{L} = \frac{- 1}{m _{P T}} = \frac{1}{3}$ .
Equation of $L$ : $y = \frac{1}{3} x + c \Rightarrow x - 3 y + λ = 0$ .

Tangency Condition for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L

Circle $C_{2} : (x - 3)^{2} + y^{2} = 1$ with center $C_{2} (3, 0)$ and radius $r_{2} = 1$ .
Distance from $(3, 0)$ to $x - 3 y + λ = 0$ is $r_{2}$ .
$\frac{∣3 - 3 ( 0 ) + λ ∣}{1 ^{2} + ( - 3 ) ^{2}} = 1$ .

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } λ

$\frac{∣3 + λ ∣}{2} = 1 \Rightarrow ∣3 + λ ∣ = 2$ .
Case 1: $3 + λ = 2 \Rightarrow λ = - 1$ .
Case 2: $3 + λ = - 2 \Rightarrow λ = - 5$ .
Possible equations: $x - 3 y = 1$ or $x - 3 y = 5$ .

Relationship Between Circles

Distance between centers $C_{1} (0, 0)$ and $C_{2} (3, 0)$ is $d = 3$ .
Sum of radii $r_{1} + r_{2} = 2 + 1 = 3$ .
Since $d = r_{1} + r_{2}$ , the circles touch externally at $(2, 0)$ .

External Center of Similitude

External center of similitude $S$ divides $C_{1} C_{2}$ in ratio $r_{1} : r_{2} = 2 : 1$ externally.
$S = (\frac{2 ( 3 ) - 1 ( 0 )}{2 - 1}, \frac{2 ( 0 ) - 1 ( 0 )}{2 - 1}) = (6, 0)$ .

Common Tangent Equation

Let the common tangent be $y - 0 = m (x - 6) \Rightarrow m x - y - 6 m = 0$ .
Distance from $C_{1} (0, 0)$ to this line is $r_{1} = 2$ .
$\frac{∣ m ( 0 ) - 0 - 6 m ∣}{m ^{2} + ( - 1 ) ^{2}} = 2$ .

Solving for Slope .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m

$\frac{6∣ m ∣}{m ^{2} + 1} = 2 \Rightarrow 3∣ m ∣ = m^{2} + 1$ .
Squaring both sides: $9 m^{2} = m^{2} + 1 \Rightarrow 8 m^{2} = 1$ .
$m = \pm \frac{1}{8} = \pm \frac{1}{2 2}$ .

Final Result

Using $m = - \frac{1}{2 2}$ : $y = - \frac{1}{2 2} (x - 6)$ .
$22 y = - x + 6 \Rightarrow x + 22 y = 6$ .
Key Takeaway: Common tangents pass through centers of similitude.
Next Challenge: Can you find the third common tangent (the internal one)?

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Conceptually Similar Problems

MathematicsEquation of Tangent and NormalJEE Advanced 1999Difficult

View in:English Hindi

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.

Answer:(x-4)^2 + y^2 = 3^2 \text{ and } (x + 4/3)^2 + y^2 = (1/3)^2; y = \pm \frac{5}{\sqrt{39}}(x + 4/5)

MathematicsEquation of Tangent and NormalJEE Advanced 2001Moderate

View in:English Hindi

The equation of the common tangent touching the circle $(x - 3)^{2} + y^{2} = 9$ and the parabola $y^{2} = 4 x$ above the x-axis is

(A)

3 y = 3 x + 1

(B)

3 y = - (x + 3)

(C)

3 y = x + 3

(D)

3 y = - (3 x + 1)

Answer:C

MathematicsEquation of Tangent and NormalModerate

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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

Answer:D

MathematicsEquation of Tangent and NormalJEE Advanced 2010Difficult

View in:English Hindi

Comprehension Passage

Tangents are drawn from the point

P (3, 4)

to the ellipse

\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1

touching the ellipse at points

A

and

B

Question 1:

The coordinates of $A$ and $B$ are

(A)

(3, 0)

and

(0, 2)

(B)

(\frac{8}{5}, \frac{2 161}{15})

and

(\frac{9}{5}, \frac{8}{5})

(C)

(- \frac{8}{5}, \frac{2 161}{15})

and

(0, 2)

(D)

(3, 0)

and

(- \frac{9}{5}, \frac{8}{5})

Answer:D

Question 2:

The orthocenter of the triangle $P A B$ is

(A)

(5, \frac{8}{7})

(B)

(\frac{7}{5}, \frac{25}{8})

(C)

(\frac{11}{5}, \frac{8}{5})

(D)

(\frac{8}{25}, \frac{7}{5})

Answer:C

Question 3:

The equation of the locus of the point whose distances from the point $P$ and the line $A B$ are equal, is

(A)

9 x^{2} + y^{2} - 6 x y - 54 x - 62 y + 241 = 0

(B)

x^{2} + 9 y^{2} + 6 x y - 54 x + 62 y - 241 = 0

(C)

9 x^{2} + 9 y^{2} - 6 x y - 54 x - 62 y - 241 = 0

(D)

x^{2} + y^{2} - 2 x y + 27 x + 31 y - 120 = 0

Answer:A

MathematicsLength of Tangent and Chord of ContactEasy

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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

Answer:C

MathematicsEquation of Tangent and NormalJEE Advanced 1991Moderate

View in:English Hindi

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.

Answer:x^2 + y^2 + 6x + 2y - 15 = 0 \text{ and } x^2 + y^2 - 10x - 10y + 25 = 0

MathematicsStandard and General Equation of a CircleJEE Main 2004Easy

View in:English Hindi

If the lines $2 x + 3 y + 1 = 0$ and $3 x - y - 4 = 0$ lie along diameter of a circle of circumference $10 π$ , then the equation of the circle is

(A)

x^{2} + y^{2} + 2 x - 2 y - 23 = 0

(B)

x^{2} + y^{2} - 2 x - 2 y - 23 = 0

(C)

x^{2} + y^{2} + 2 x + 2 y - 23 = 0

(D)

x^{2} + y^{2} - 2 x + 2 y - 23 = 0

Answer:D

MathematicsIntercepts Made by a CircleJEE Advanced 1999Moderate

View in:English Hindi

Let $L_{1}$ be a straight line passing through the origin and $L_{2}$ be the straight line $x + y = 1$ . If the intercepts made by the circle $x^{2} + y^{2} - x + 3 y = 0$ on $L_{1}$ and $L_{2}$ are equal, then which of the following equations can represent $L_{1}$ ?

* Multiple Correct Options

(A)

x + y = 0

(B)

x - y = 0

(C)

x + 7 y = 0

(D)

x - 7 y = 0

Answer:B, C

MathematicsDirector Circle and Family of CirclesModerate

View in:English Hindi

Two circles $x^{2} + y^{2} = 6$ and $x^{2} + y^{2} - 6 x + 8 = 0$ are given. Then the equation of the circle through their points of intersection and the point $(1, 1)$ is

(A)

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

(B)

x^{2} + y^{2} - 3 x + 1 = 0

(C)

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

(D)

none of these

Answer:B

MathematicsCondition for OrthogonalityJEE Main 2005Moderate

View in:English Hindi

If a circle passes through the point $(a, b)$ and cuts the circle $x^{2} + y^{2} = p^{2}$ orthogonally, then the equation of the locus of its centre is

(A)

x^{2} + y^{2} - 3 a x - 4 b y + (a^{2} + b^{2} - p^{2}) = 0

(B)

2 a x + 2 b y - (a^{2} + b^{2} + p^{2}) = 0

(C)

x^{2} + y^{2} - 2 a x - 3 b y + (a^{2} - b^{2} - p^{2}) = 0

(D)

2 a x + 2 b y - (a^{2} + b^{2} + p^{2}) = 0

Answer:D

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A possible equation of L is

A common tangent of the two circles is

Visualized Solution (English)

Equation of Tangent .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } PT

Defining Line .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L

Tangency Condition for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } λ

Relationship Between Circles

External Center of Similitude

Common Tangent Equation

Solving for Slope .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m

Final Result

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the common tangent touching the circle (x−3)2+y2=9 and the parabola y2=4x above the x-axis is

.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

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coordinates of A and B are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The orthocenter of the triangle PAB is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the locus of the point whose distances from the point P and the line AB are equal, 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; } 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.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 2x+3y+1=0 and 3x−y−4=0 lie along diameter of a circle of circumference 10π, then the equation of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two circles x2+y2=6 and x2+y2−6x+8=0 are given. Then the equation of the circle through their points of intersection and the point (1,1) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a circle passes through the point (a,b) and cuts the circle x2+y2=p2 orthogonally, then the equation of the locus of its centre is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A possible equation of L is

A common tangent of the two circles is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the common tangent touching the circle (x−3)2+y2=9 and the parabola y2=4x above the x-axis is

.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

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coordinates of A and B are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The orthocenter of the triangle PAB is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the locus of the point whose distances from the point P and the line AB are equal, 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; } 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.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 2x+3y+1=0 and 3x−y−4=0 lie along diameter of a circle of circumference 10π, then the equation of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two circles x2+y2=6 and x2+y2−6x+8=0 are given. Then the equation of the circle through their points of intersection and the point (1,1) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a circle passes through the point (a,b) and cuts the circle x2+y2=p2 orthogonally, then the equation of the locus of its centre is

A possible equation of $L$ is

Equation of Tangent $P T$

Defining Line $L$

Tangency Condition for $L$

Solving for $λ$

Solving for Slope $m$

The equation of the common tangent touching the circle $(x - 3)^{2} + y^{2} = 9$ and the parabola $y^{2} = 4 x$ above the x-axis is

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 coordinates of $A$ and $B$ are

The orthocenter of the triangle $P A B$ is

The equation of the locus of the point whose distances from the point $P$ and the line $A B$ are equal, 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

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.

If the lines $2 x + 3 y + 1 = 0$ and $3 x - y - 4 = 0$ lie along diameter of a circle of circumference $10 π$ , then the equation of the circle is

Two circles $x^{2} + y^{2} = 6$ and $x^{2} + y^{2} - 6 x + 8 = 0$ are given. Then the equation of the circle through their points of intersection and the point $(1, 1)$ is

If a circle passes through the point $(a, b)$ and cuts the circle $x^{2} + y^{2} = p^{2}$ orthogonally, then the equation of the locus of its centre is

A possible equation of $L$ is

The equation of the common tangent touching the circle $(x - 3)^{2} + y^{2} = 9$ and the parabola $y^{2} = 4 x$ above the x-axis is

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 coordinates of $A$ and $B$ are

The orthocenter of the triangle $P A B$ is

The equation of the locus of the point whose distances from the point $P$ and the line $A B$ are equal, 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

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.

If the lines $2 x + 3 y + 1 = 0$ and $3 x - y - 4 = 0$ lie along diameter of a circle of circumference $10 π$ , then the equation of the circle is

Two circles $x^{2} + y^{2} = 6$ and $x^{2} + y^{2} - 6 x + 8 = 0$ are given. Then the equation of the circle through their points of intersection and the point $(1, 1)$ is

If a circle passes through the point $(a, b)$ and cuts the circle $x^{2} + y^{2} = p^{2}$ orthogonally, then the equation of the locus of its centre is