MyQuestionIcon

1

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

Point Form of Tangent: Ellipse

The lengths o...

Question

The lengths of tangents from a fixed point to three circles of a co-axial system are $t_{1}, t_{2}, t_{3}$ respectively. If $P, Q, R$ be the centres of the three circles, show that $Q R t_{1}^{2} + R P t_{2}^{2} + P Q t_{3}^{2} = 0$ .

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 $\sum g_{1} (g_{2} - g_{3}) = 0$ .

Suggest Corrections

thumbs-up

0

Similar questions

Q. The lengths of tangents from a fixed point to three circles of coaxial system are

t_{1}, t_{2}, t_{3}

and if P, Q and R be the centres, then

Q R t_{1}^{2} + R P t_{2}^{2} + P Q t_{3}^{2}

P, Q, R, S

are the centres of the four circles each of which is cut by a fixed circle orthogonally. If

t_{1}^{2}, t_{2}^{2}, t_{3}^{2}, t_{4}^{2}

be the squares of the lengths of tangents to the four circles from a point in their plane, then prove that

t_{1}^{2} △ Q R S + t_{2}^{2} △ R S P + t_{3}^{2} △ S P Q + t_{4}^{2} △ P Q R = 0

Q. What is the general equation of a circle passing through two given points

(x_{1}, y_{1})

(x_{2}, y_{2})

S_{1}, S_{2}, S_{3}

be three members of this family and

t_{2}, t_{3}

be the tangents from any point on

S_{1}

S_{2}

S_{3}

, then show that

t_{2} / t_{3}

S_{1} = 0, S_{2} = 0

S_{3} = 0

are the three circles whose radical centre is the point

P

, then the lengths

l_{1}, l_{2}, l_{3}

of the tangent from

P

to the three circles are such that

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?

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Line and Ellipse

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Point Form of Tangent: Ellipse

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon