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Equation of a Chord Joining Two Points with Circle in Parametric Form

Let S be the ...

Question

Let S be the circle in the xy-plane defined by the equation $x^{2} + y^{2} = 4$ . Let $E_{1} E_{2}$ and $F_{1} F_{2}$ be the chord of S passing through the point $P_{o} (1,$ and parallel to the x-axis and the y-axis, respectively. Let $G_{1} G_{2}$ be the chord of S passing through $P_{o}$ and having slop- $1$ . Let the tangents to S at $E_{1}$ and $E_{2}$ meet at $E_{3}$ , the tangents of S at $F_{1}$ and $F_{2}$ meet at $F_{3}$ , and the tangents to S at $G_{1}$ and $G_{2}$ meet at $G_{3}$ . Then, the points $E_{3}$ , $F_{3}$ and $G_{3}$ lie on the curve.

x + y = 4

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(x - 4)^{2} + (y - 4)^{2} = 16

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(x - 4) (y - 4) = 4

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x y = 4

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Solution

The correct option is D $x + y = 4$
Co-ordinates of $E_{1}$ and $E_{2}$ are obtained by solving $y = 1$ and $x^{2} + y^{2} = 4$
$∴ E_{1} (- \sqrt{3}, 1)$ and $E_{2} (\sqrt{3}, 1)$
co-ordinates of $F_{1}$ and $F_{2}$ are obtained by solving
$x = 1$ and $x^{2} + y^{2} = 4$
$F_{1} (1, \sqrt{3})$ and $F_{2} (1, - \sqrt{3})$
Tangent at $E_{1} : - \sqrt{3} x + y = 4$
Tangent at $E_{2} : \sqrt{3} x + y = 4$
$∴ E_{3} (0, 4)$
Tangent at $F_{1} : x + \sqrt{3} y = 4$
Tangent at $F_{2} : x - \sqrt{3} y = 4$
$∴ F_{3} (4, 0)$
and similarly $G_{3} (2, 2)$
$(0, 4), (4, 0)$ and $(2, 2)$ lies on $x + y = 4$

Suggest Corrections

Similar questions

Q. Let

S

be the circle in the

x y

-plane defined by the equation

x^{2} + y^{2} = 4

.

Let

E_{1} E_{2}

and

F_{1} F_{2}

be the chords of

S

passing through the point

P_{0} (1, 1)

and parallel to the

x

-axis and the

y

-axis, respectively. Let

G_{1} G_{2}

be the chord of

S

passing through

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and having slope

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. Let the tangents to

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E_{1}

and

E_{2}

meet at

E_{3}

, the tangents to

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F_{1}

and

F_{2}

meet at

F_{3}

, and the tangents to

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G_{1}

and

G_{2}

meet at

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. Then, the points

E_{3}, F_{3}

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G_{3}

lie on the curve

Q. Let

S

be the circle in the

x y

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x^{2} + y^{2} = 4

.

Let

E_{1} E_{2}

and

F_{1} F_{2}

be the chords of

S

passing through the point

P_{0} (1, 1)

and parallel to the

x

-axis and the

y

-axis, respectively. Let

G_{1} G_{2}

be the chord of

S

passing through

P_{0}

and having slope

- 1

. Let the tangents to

S

E_{1}

and

E_{2}

meet at

E_{3}

, the tangents to

S

F_{1}

and

F_{2}

meet at

F_{3}

, and the tangents to

S

G_{1}

and

G_{2}

meet at

G_{3}

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E_{3}, F_{3}

, and

G_{3}

lie on the curve

Q. Let S be the circle in the xy-plane defined by the equation

x^{2} + y^{2} = 4

. Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the segment MN must lie on the curve.

Q. Let

R S

be the diameter of the circle

x^{2} + y^{2} = 1

, where

S

is the point

(1, 0)

. Let

P

be a variable point (other than

R

and

S

) on the circle and tangents to the circle at

S

and

P

meet at the point

Q

. The normal to the circle at

P

intersects a line drawn through

Q

parallel to

R S

at point

E

. Then the locus of

E

passes through the point(s)

Q. Let

E_{1}

and

E_{2}

be the two ellipses centred at origin. The major axis of

E_{1}

and

E_{2}

lie along the

x -

axis and

y -

axis respectively. Let

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be the circle

x^{2} + (y - 1)^{2} = 2,

the straight line

x + y = 3

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S, E_{1}

and

E_{2}

P, Q

and

R

respectively such that

P Q = P R = \frac{2 \sqrt{2}}{3} .

e_{1}

and

e_{2}

are the eccentricities of

E_{1}

and

E_{2},

then which of the following is/are correct

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