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Position of a Point with Respect to Circle

Let T be th...

Question

Let $T$ be the line passing through the points $P (- 2, 7)$ and $Q (2, - 5)$ . Let $F_{1}$ be the set of all pairs of circles $(S_{1}, S_{2})$ such that $T$ is tangent to $S_{1}$ at $P$ and tangent to $S_{2}$ at $Q$ , and also such that $S_{1}$ and $S_{2}$ touch each other at a point, say, $M$ . Let $E_{1}$ be the set representing the locus of $M$ as the pair $(S_{1}, S_{2})$ varies in $F_{1}$ . Let the set of all straight line segments joining a pair of distinct points of $E_{1}$ and passing through the point $R (1, 1)$ be $F_{2}$ . Let $E_{2}$ be the set of the mid-points of the line segments in the set $F_{2}$ . Then, which of the following statement(s) is (are) TRUE?

The point

(- 2, 7)

lies in

E_{1}

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The point

(\frac{4}{5}, \frac{7}{5})

does NOT lie in

E_{2}

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The point

(\frac{1}{2}, 1)

lies in

E_{2}

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The point

(0, \frac{3}{2})

doe NOT lie in

E_{1}

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Solution

The correct option is D The point $(0, \frac{3}{2})$ doe NOT lie in $E_{1}$
$A P = A Q = A M$
Locus of M is a circle having $P Q$ as its diameter
Hence, $E_{1} : (x - 2) (x + 2) + (y - 7) (y + 5) = 0$ and $x \neq \pm 2$
Locus of $B$ (midpoint)
is a circle having $R C$ as its diameter
$E_{2} | : x (x - 1) + (y - 1)^{2} = 0$
Now, after checking the options, we get $(D)$ .

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Similar questions

Q. Let

T

be the line passing through the points

P (- 2, 7)

and

Q (2, - 5)

. Let

F_{1}

be the set of all pairs of circles

(S_{1}, S_{2})

such that

T

is tangent to

S_{1}

P

and tangent to

S_{2}

Q

, and also such that

S_{1}

and

S_{2}

touch each other at a point, say,

M

. Let

E_{1}

be the set representing the locus of

M

as the pair

(S_{1}, S_{2})

varies in

F_{1}

. Let the set of all straight line segments joining a pair of distinct points of

E_{1}

and passing through the point

R (1, 1)

F_{2}

. Let

E_{2}

be the set of the mid-points of the line segments in the set

F_{2}

. Then, which of the following statement(s) is (are) TRUE?

Q. Consider three sets

E_{1} = {1, 2, 3},

F_{1} = {1, 3, 4}

and

G_{1} = {2, 3, 4, 5} .

Two elements are chosen at random, without replacement, from the set

E_{1},

and let

S_{1}

denote the set of these chosen elements. Let

E_{2} = E_{1} - S_{1}

and

F_{2} = F_{1} \cup S_{1} .

Now two elements are chosen at random, without replacement, from the set

F_{2}

and let

S_{2}

denote the set of these chosen elements.

Let

G_{2} = G_{1} \cup S_{2} .

Finally, two elements are chosen at random, without replacement from the set

G_{2}

and let

S_{3}

denote the set of these chosen elements.
Let

E_{3} = E_{2} \cup S_{3} .

Given that

E_{1} = E_{3},

let

p

be the conditional probability of the event

S_{1} = {1, 2} .

Then the value of

p

S_{1}

and

S_{2}

are two sources of light which produce individually disturbance at point

P

given by

{\to E}_{1} = 3 sin ω t

and

{\to E}_{2} = 4 cos ω t

. Assuming

_{1}

and

_{2}

to be along the same line, the resultant of their super position is

Q. Let

E_{1} : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, a > b .

Let

E_{2}

be another ellipse such that it touches the end points of major axis of

E_{1}

and the foci of

E_{2}

are the end points of minor axis of

E_{1} .

E_{1}

and

E_{2}

have same eccentricities, then its value is

Q. Let

E_{1}

and

E_{2}

be two ellipses

\frac{x^{2}}{a^{2}} + y^{2} = 1

and

x^{2} + \frac{y^{2}}{a^{2}} = 1

(where a is parameter) the locus of points of intersection of the ellipses

E_{1}

and

E_{2}

is a set of curves

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The correct option is D The point (0,32) doe NOT lie in E1AP=AQ=AMLocus of M is a circle having PQ as its diameterHence, E1:(x−2)(x+2)+(y−7)(y+5)=0 and x≠±2Locus of B (midpoint)is a circle having RC as its diameterE2|:x(x−1)+(y−1)2=0Now, after checking the options, we get (D).