C₁ and C₂ are two concentric circles,...

MathematicsEquation of Tangent and NormalJEE Advanced 1998Moderate

$C_{1}$ and $C_{2}$ are two concentric circles, the radius of $C_{2}$ being twice that of $C_{1}$ . From a point $P$ on $C_{2}$ , tangents $P A$ and $P B$ are drawn to $C_{1}$ . Prove that the centroid of the triangle $P A B$ lies on $C_{1}$ .

Visualized Solution (English)

Setting up the Circles .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C_{1} and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C_{2}

Let the center of both circles be the origin $(0, 0)$ .
Equation of $C_{1}$ : $x^{2} + y^{2} = r^{2}$
Equation of $C_{2}$ : $x^{2} + y^{2} = (2 r)^{2} = 4 r^{2}$

Defining Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P on .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C_{2}

Let point $P$ lie on the x-axis for simplicity.
Since $P$ is on $C_{2}$ , its coordinates are $(2 r, 0)$ .

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

Tangents $P A$ and $P B$ are drawn to $C_{1}$ .
The line $A B$ is the Chord of Contact of $P$ with respect to $C_{1}$ .
Equation of chord of contact is $T = 0$ : $x x_{1} + y y_{1} = r^{2}$ .

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

Substitute $P (2 r, 0)$ into the formula:
$x (2 r) + y (0) = r^{2}$
$2 r x = r^{2} ⟹ x = \frac{r}{2}$

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

Points $A$ and $B$ lie on $C_{1}$ and the line $x = \frac{r}{2}$ .
Substitute $x = \frac{r}{2}$ into $x^{2} + y^{2} = r^{2}$ :
$(\frac{r}{2})^{2} + y^{2} = r^{2}$
$\frac{r ^{2}}{4} + y^{2} = r^{2} ⟹ y^{2} = \frac{3 r ^{2}}{4}$

Coordinates of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B (Final)

Solving for $y$ : $y = \pm \frac{r 3}{2}$
Point $A = (\frac{r}{2}, \frac{r 3}{2})$
Point $B = (\frac{r}{2}, - \frac{r 3}{2})$

Calculating the Centroid .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } G

Centroid $G (x_{G}, y_{G})$ of $△ P A B$ is given by:
$x_{G} = \frac{x _{P} + x _{A} + x _{B}}{3}$
$x_{G} = \frac{2 r + \frac{r}{2} + \frac{r}{2}}{3} = \frac{3 r}{3} = r$

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y -coordinate of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } G

$y_{G} = \frac{y _{P} + y _{A} + y _{B}}{3}$
$y_{G} = \frac{0 + \frac{r 3}{2} - \frac{r 3}{2}}{3} = 0$
Centroid $G = (r, 0)$

Final Verification

Check if $G (r, 0)$ satisfies $x^{2} + y^{2} = r^{2}$ :
LHS: $r^{2} + 0^{2} = r^{2}$
RHS: $r^{2}$
LHS = RHS. Therefore, $G$ lies on $C_{1}$ . Hence Proved.

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$C_{1}$ and $C_{2}$ are two concentric circles, the radius of $C_{2}$ being twice that of $C_{1}$ . From a point $P$ on $C_{2}$ , tangents $P A$ and $P B$ are drawn to $C_{1}$ . Prove that the centroid of the triangle $P A B$ lies on $C_{1}$ .

Visualized Solution (English)

Setting up the Circles $C_{1}$ and $C_{2}$

Defining Point $P$ on $C_{2}$

Chord of Contact $A B$

Equation of Line $A B$

Finding Coordinates of $A$ and $B$

Coordinates of $A$ and $B$ (Final)

Calculating the Centroid $G$

Calculating $y$ -coordinate of $G$

Final Verification

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Visualized Solution (English)

Setting up the Circles .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C1​ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Defining Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P on .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Chord of Contact .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AB

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

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

Coordinates of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B (Final)

Calculating the Centroid .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } G

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y-coordinate of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } G

Final Verification

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a vertex of a triangle is (1,1) and the mid points of two sides through this vertex are (−1,2) and (3,2) then the centroid of the triangle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let AB be a chord of the circle x2+y2=r2 subtending a right angle at the centre. Then the locus of the centroid of the triangle PAB as P moves on the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a vertex of a triangle is (1,1) and the mid points of two sides through this vertex are (−1,2) and (3,2) then the centroid of the triangle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let AB be a chord of the circle x2+y2=r2 subtending a right angle at the centre. Then the locus of the centroid of the triangle PAB as P moves on the circle is

Setting up the Circles $C_{1}$ and $C_{2}$

Defining Point $P$ on $C_{2}$

Chord of Contact $A B$

Equation of Line $A B$

Finding Coordinates of $A$ and $B$

Coordinates of $A$ and $B$ (Final)

Calculating the Centroid $G$

Calculating $y$ -coordinate of $G$

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