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Standard XII

Mathematics

Condition of Concurrency of 3 Straight Lines

PM and PN are...

Question

PM and PN are the perpendiculars from any point P on the rectangular hyperbola xy = 8 to the asymptotes. If the locus of the mid point of MN is a conic, then the least distance of (1, 1) to director circle of the conic is

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Solution

The correct option is B

OMPN is rectangle.

$P = (C t, \frac{c}{t})$

$M i d p o i n t = (\frac{c t}{2}, \frac{c}{2 t}) = (x, y) ∴ c y = \frac{c^{2}}{4} \Rightarrow e = \sqrt{2}$

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Similar questions

P M

and

P N

are the perpendiculars from any point

P

on the rectangular hyperbola

x y = 8

to the asymptotes. If the locus of the mid point of MN is a conic, then the least distance of

(1, 1)

to director circle of the conic is

P M

and

P N

are the perpendiculars from any point

P

on the rectangular hyperbola

x y = 8

to the asymptotes. If the locus of the mid point of MN is a conic, then the least distance of

(1, 1)

to director circle of the conic is

PM and PN are the perpendiculars from any point P on the rectangular hyperbola $x y = c^{2}$ to the asymptotes.

If the locus of the mid point of MN is a conic, then its eccentricity is

Q. If

P N

is the perpendicular from a point

P

on a rectangular hyperbola to its asymptotes, the locus of the mid-point of

P N

A_{1}

and

A_{2}

are the vertices of the conic

C_{1} : 4 (x - 3)^{2} + 9 (y - 2)^{2} - 36 = 0

and point

P

is moving in the plane such that

| P A_{1} - P A_{2} | = 3 \sqrt{2} .

Then the locus of

P

is another conic

C_{2} .

D_{1}

denotes the distance between foci of conic

C_{2},

D_{2}

denotes the product of perpendicular distances from the points

A_{1}

and

A_{2}

upon any tangent drawn to the conic

C_{2}

and

D_{3}

denotes length of tangent drawn from any point on the auxiliary circle of conic

C_{1}

to the auxiliary circle of the conic

C_{2},

then

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

is equal to

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PM and PN are the perpendiculars from any point P on the rectangular hyperbola xy = 8 to the asymptotes. If the locus of the mid point of MN is a conic, then the least distance of (1, 1) to director circle of the conic is

The correct option is B OMPN is rectangle. P=(Ct,ct) Mid point=(ct2,c2t)=(x,y) ∴ cy=c24⇒e=√2

The correct option is B

OMPN is rectangle.

$P = (C t, \frac{c}{t})$

$M i d p o i n t = (\frac{c t}{2}, \frac{c}{2 t}) = (x, y) ∴ c y = \frac{c^{2}}{4} \Rightarrow e = \sqrt{2}$