CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Tangent of a Curve y =f(x)

P, Q, R are t...

Question

$P, Q, R$ are the centres and $r_{1}, r_{2}, r_{3}$ are the radii respectively of three coaxial circles, show that $r_{1}^{2} Q R + r_{2}^{2} R P + r_{3}^{2} P Q = P Q . Q R . R P$ .

Open in App

Solution

By special choice of axes the equation of coaxial system of circles can be put in the form
$x^{2} + y^{2} + 2 g_{r} x + c = 0$ where $r = 1, 2, 3$ .
The co-ordinates of the centres of the three circles are
$P (- g_{1}, 0), Q (- g_{2}, 0), R (- g_{3}, 0)$
$∴ Q R = (- g_{3}) - (- g_{2}) = g_{2} - g_{3}$
$R P = g_{3} - g_{1}$ and $P Q = g_{1} - g_{2}$ .
If the fixed point from where the tangents be drawn be $(h, k)$ , then
$t_{1}^{2} = h^{2} + k^{2} + 2 g_{1} h + c$ etc.
$∴ Q R t_{1}^{2} + R P t_{2}^{2} + P Q t_{3}^{2}$
$= (g_{2} - g_{3}) {h^{2} + k^{2} + c + 2 g_{1} h} + . . . . + . . . . = 0$
or $(h^{2} + k^{2} + c) \sum (g_{2} - g_{3}) + 2 h \sum g_{2} (g_{1} - g_{3}) = 0$
$∵ \sum (g_{2} - g_{3}) = 0$ and $\sum g_{1} (g_{2} - g_{3}) = 0$ .

From part (a), $r_{1}^{2} = (g_{1})^{2} - c$ etc.
$∴ r_{1}^{2} . Q R + r_{2}^{2} . R P + r_{3}^{2} . P Q$
$= (g_{1}^{2} - c) (g_{2} - g_{3}) + (g_{2}^{2} - c) (g_{3} - g_{1}) + (g_{3}^{2} - c) (g_{1} - g_{2})$
$= \sum g_{1}^{2} (g_{2} - g_{3}) - c \sum (g_{2} - g_{3})$
$= - (g_{1} - g_{2}) (g_{2} - g_{3}) (g_{3} - g_{1}) - c . 0$
$= - P Q . Q R . R P$ .

Suggest Corrections

thumbs-up

0

Similar questions

Q. Three Circles with centres

C_{1}, C_{2}, C_{3}

r_{1}, r_{2}, r_{3}

r_{1} < r_{2} < r_{3}

are placed as shown in the figure.
Then,

r_{2} =

Q. Three circles

C_{1}, C_{2}

C_{3}

r_{1}, r_{2}

r_{1} + r_{2}

respectively are such that

C_{1}

C_{2}

touch each other externally and

C_{3}

internally. Another circle with radius

r_{3}

touches all the three circles. If

r_{1} > r_{2} > r_{3}

r_{1}, r_{2}, r_{3}

are in A.P. then

{(\frac{r_{1}}{r_{2}})}^{3}

Q. Three circles of radii

r_{1}

r_{2}

r_{3}

are drawn concentric to each other. The radii

r_{1}

r_{2}

are such that the area of the circle with radius

r_{1}

is equal to the area between the circles of radii

r_{2}

r_{1}

. The area between the circles of radii

r_{3}

r_{2}

is equal to the area between the circles of radii

r_{2}

r_{1}

What is the value of

r_{1} : r_{2} : r_{3}

?

Q. Three concentric circles with centre O and radii

r_{1}

r_{2}

r_{3}

r_{1}

r_{2}

, P is a point on the circle with radius

r_{2}

and Q is a point on circle with radius r3 how many tangents can be drawn from P and Q to the circles?

A B C

be a triangle with in centre

I

r

D, E, F

be the feet of the perpendiculars from

I

B C, C A

A B

respectively. If

r_{1}, r_{2} r_{3}

are the radii of circles inscribed in then prove

\frac{r_{1}}{r - r_{1}}_\frac{r_{2}}{r - r_{2}} + \frac{r_{3}}{r - r_{3}} = \frac{r_{1} r_{2} r_{3}}{(r - r_{1}) (r - r_{2}) (r - r_{3})}

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Characteristics of Sound Waves

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Tangent of a Curve y =f(x)

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon