JEE Advanced 2012

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Animated Solution for Mathematics - Circles: 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

Q: Why do we equate the coefficients? Is that always valid?

Yes, if two equations $Ax + By + C = 0$ and $A'x + B'y + C' = 0$ represent the same line, then $A/A' = B/B' = C/C'$. It's a fundamental property of linear equations in coordinate geometry.

Select Answer:

Visualized Solution

Visualizing the Setup

Circle: $x^{2} + y^{2} = 9$ (Center $(0, 0)$ , Radius $3$ )
Line: $4 x - 5 y = 20$

Defining Point $P$ on the Line

Let $P (x_{1}, y_{1})$ be any point lying on the given line $4 x - 5 y = 20$ .
Since $P$ lies on this line, its coordinates must satisfy the line's equation:
$4 x_{1} - 5 y_{1} = 20$

Equation of Chord of Contact

The equation of the chord of contact of tangents drawn from an external point $P (x_{1}, y_{1})$ to the circle $x^{2} + y^{2} = a^{2}$ is given by $T = 0$ .
For our circle $x^{2} + y^{2} = 9$ , the equation becomes:
$x x_{1} + y y_{1} = 9$

Introducing the Midpoint $M (h, k)$

Let $M (h, k)$ be the midpoint of this chord of contact.
The equation of a chord of a circle with a given midpoint $(h, k)$ is represented by $T = S_{1}$ .
For our circle, this gives:
$x h + y k - 9 = h^{2} + k^{2} - 9$
Simplifying: $x h + y k = h^{2} + k^{2}$

Comparing the Two Equations

Both equations represent the exact same physical chord of contact:
Equation 1: $x x_{1} + y y_{1} = 9$
Equation 2: $x h + y k = h^{2} + k^{2}$
Therefore, their corresponding coefficients must be proportional:
$\frac{x _{1}}{h} = \frac{y _{1}}{k} = \frac{9}{h ^{2} + k ^{2}}$

Expressing $x_{1}$ and $y_{1}$

From the ratio $\frac{x _{1}}{h} = \frac{9}{h ^{2} + k ^{2}}$ , we get:
$x_{1} = \frac{9 h}{h ^{2} + k ^{2}}$
From the ratio $\frac{y _{1}}{k} = \frac{9}{h ^{2} + k ^{2}}$ , we get:
$y_{1} = \frac{9 k}{h ^{2} + k ^{2}}$

Substituting into the Line Constraint

Recall the constraint for point $P (x_{1}, y_{1})$ lying on the line:
$4 x_{1} - 5 y_{1} = 20$
Substitute the expressions for $x_{1}$ and $y_{1}$ :
$4 (\frac{9 h}{h ^{2} + k ^{2}}) - 5 (\frac{9 k}{h ^{2} + k ^{2}}) = 20$

Simplifying the Equation

Combine the fractions over the common denominator $h^{2} + k^{2}$ :
$\frac{36 h - 45 k}{h ^{2} + k ^{2}} = 20$
Multiply both sides by $h^{2} + k^{2}$ :
$36 h - 45 k = 20 (h^{2} + k^{2})$
Rearranging terms:
$20 (h^{2} + k^{2}) - 36 h + 45 k = 0$

Finding the Final Locus

To find the locus of the midpoint $M (h, k)$ , replace $h$ with $x$ and $k$ with $y$ :
$20 (x^{2} + y^{2}) - 36x + 45y = 0$
This is the equation of a circle, which represents the locus of the midpoint.
Correct Option: (1)

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The Sigma Insight: Length of Tangent and Chord of Contact

Solution Diagram

Analyzing the Setup

We are given a circle defined by the equation

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

, which has a center at the origin

(0, 0)

and a radius of

3

. A line is defined by the equation

4 x - 5 y = 20

We consider a point

P (x_{1}, y_{1})

that lies on this line. From

P

, two tangents are drawn to the circle, forming a chord of contact. We seek the locus of the midpoint

M (h, k)

of this chord as

P

moves along the line.

The Dual Nature of the Chord

The equation of the chord of contact for a point

P (x_{1}, y_{1})

with respect to the circle

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

is given by

T = 0

. This yields:

x x_{1} + y y_{1} = 9

Alternatively, the equation of a chord of a circle with a given midpoint

M (h, k)

is given by

T = S_{1}

. Substituting the coordinates of

M

into the circle equation, we obtain:

x h + y k - 9 = h^{2} + k^{2} - 9

Simplifying this expression, the constant terms cancel out to provide the second representation of the chord:

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

The Mathematical Bridge

Since both equations represent the same line, their coefficients must be proportional. We establish the following relationship:

\frac{x _{1}}{h} = \frac{y _{1}}{k} = \frac{9}{h ^{2} + k ^{2}}

From these ratios, we can express the coordinates of the external point

P (x_{1}, y_{1})

in terms of the midpoint coordinates

(h, k)

x_{1} = \frac{9 h}{h ^{2} + k ^{2}}, y_{1} = \frac{9 k}{h ^{2} + k ^{2}}

The Final Transformation

We know that point

P (x_{1}, y_{1})

must satisfy the constraint of the line

4 x - 5 y = 20

. Substituting our expressions for

x_{1}

and

y_{1}

into this equation, we get:

4 (\frac{9 h}{h ^{2} + k ^{2}}) - 5 (\frac{9 k}{h ^{2} + k ^{2}}) = 20

Multiplying through by the denominator

(h^{2} + k^{2})

, we obtain:

36 h - 45 k = 20 (h^{2} + k^{2})

Replacing

h

and

k

with the general variables

x

and

y

, we arrive at the final equation for the locus:

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

Common Traps & FAQs

Why can't we just use the distance formula to find the length of the chord?

The distance formula gives you the length, but the locus problem asks for the *path* of the midpoint. We need the equation of the line, not just its length. The locus is the set of all possible midpoints as the point

P

moves along the line.

Why do we equate the coefficients? Is that always valid?

Yes, if two equations

A x + B y + C = 0

and

A^{'} x + B^{'} y + C^{'} = 0

represent the same line, then

A / A^{'} = B / B^{'} = C / C^{'}

. It's a fundamental property of linear equations in coordinate geometry.

Why is the locus a circle? I thought it would be a line.

The locus is the set of all possible midpoints. As the point

P

moves along the line, the chord changes, and its midpoint moves. The resulting path is a circle, which is a beautiful, non-linear result that emerges from the algebraic manipulation.

Animated Solution for Mathematics - Circles: 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

Select Answer:

Visualized Solution

Visualizing the Setup

Defining Point $P$ on the Line

Equation of Chord of Contact

Introducing the Midpoint $M (h, k)$

Comparing the Two Equations

Expressing $x_{1}$ and $y_{1}$

Substituting into the Line Constraint

Simplifying the Equation

Finding the Final Locus

The Sigma Insight: Length of Tangent and Chord of Contact

Analyzing the Setup

The Dual Nature of the Chord

The Mathematical Bridge

The Final Transformation

Similar Questions

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Comprehension Passage

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Select Answer:

Visualized Solution

Visualizing the Setup

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

Equation of Chord of Contact

Introducing the Midpoint .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } M(h,k)

Comparing the Two Equations

Expressing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x1​ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y1​

Substituting into the Line Constraint

Simplifying the Equation

Finding the Final Locus

The Sigma Insight: Length of Tangent and Chord of Contact

Analyzing the Setup

The Dual Nature of the Chord

The Mathematical Bridge

The Final Transformation

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.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 .........

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Defining Point $P$ on the Line

Introducing the Midpoint $M (h, k)$

Expressing $x_{1}$ and $y_{1}$

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 .........

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 .........

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

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.

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

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 .........