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23 days ago

If a point P is moving such that the lengths of the tangents drawn from P to the circles

Mathematics

1 answer:

Katyanochek1 [597]23 days ago

6 0

<em>Hello ~</em>

<em />

My name is Irxdscent, and I will be answering your question today! :D

<h2>→ <u>Answer :-</u></h2>

<u />

<u /> $\sf{(x_{1} \: - y_{1})$

<h2>→ <u>Explanation :-</u></h2>

<u />

<u /> $\sf{Equations \: of \: the \: circles \: are:}$

$\sf{S = x^2 \: + \: y^2 \: - \: 4x \: - \: 6y \: - \: 12 = 0}$

$\sf{S_{1} = y^2 \: + \: y^2 \: + \: 6x \: + \: 18y \: + \: 26 = 0}$

$\sf{P(x_{1}, \: y_{1}) \: is \: any \: point \: on \: the \: locus \: and \: PT_{1,} \: PT_{2} \: are \: the \: tangents \: from \: P \: to \: the \: two \: circles.$

$\sf{Given \: condition \: is:}$

$\sf{\frac{PT_{1}}{PT_{2}} = \frac{2}{3}$

$\sf{3PT_{1} \: = \: 2PT_{2}$

$\sf{3(x^2 \: + \: y^2 \: - \: 4x \: - \: 6y \: - \: 12) = 2(x^2 \: + \: y^2 \: + \: 6x \: + \: 18y \: + 26)$

$\sf{3x^2 \: + \:3 y^2 \: - \: 12x \: - \: 18y \: - \: 36 = 2x^2 \: + \: 2y^2 \: + \: 12x \: + \: 36y \: + 52}$

$\sf{x^2 \: + \: y^2 \: - \: 24x \: - \: 54y \: - \: 88 = 0}$

$\sf{This \: is \: the \: locus \: P \: \sf{(x_{1} \: - y_{1})$

Hope It Helps! ~

<em>~ Learn With Lrxdscent ~</em>

<em />

<em /> $\underline{Answer :}$

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