A-1- 2 and B-7- 10 are two points- If Px- y is a point such that - APB-60- and area of - APB is maximum- then which of the following is are TRUE -

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Angle Subtended by an Arc of a Circle on the Circle and at the Center

A≡1, 2 and ...

Question

$A \equiv (1, 2)$ and $B \equiv (7, 10)$ are two points. If $P (x, y)$ is a point such that $∠ A P B = 60^{\circ}$ and area of $Δ A P B$ is maximum, then which of the following is (are) TRUE ?

P

lies on any line perpendicular to

A B

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P

lies on the perpendicular bisector of

A B

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P

lies on the line

3 x + 4 y = 36

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Radius of circumcircle of

Δ P A B

10

units

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Solution

The correct options are
B $P$ lies on the perpendicular bisector of $A B$
C $P$ lies on the line $3 x + 4 y = 36$
Equation of perpendicular of $A B$ is
$y - y_{1} = - m (y - y_{1})$
$y - 6 = - \frac{3}{4} (x - 4)$
$\Rightarrow 3 x + 4 y = 36$
Also, $A B = 10, A L = 5$
Therefore, $A H =$ radius of circle $= \frac{10}{\sqrt{3}} = A L c o s e c 60^{\circ}$ and $A L$ perpendicularly bisect the $A B .$

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Similar questions

Let A(1, 2) and B(7, 10) are two points. If P(x, y) is a point such that the $∠ A P B$ is $60^{\circ}$ and the area of the $△ A P B$ is maximum, then which of the following is/are true.

A (1, 2)

and

B (7, 10)

are two points. If

p (x)

is a point such that the

∠ A P B

60^{\circ}

and the area of the

△ A P B

is maximum, then which of the following is (are) true?
(A)

p

lies on the straight line

3 x + 4 y = 36

(B)

p

lies on any line perpendicular to

A B

Q. If

A (1, 1)

and

B (2, - 3)

are two points and

P (h, k)

lies on the line

A B

such that

P A = 3 A B

, where

B

lies in between

A

and

P

, then the value of

h - k

Q. In given figure

A B

is a line segment and line

l

is its perpendicular bisector. If a point

P

lies on

l

, show that

P

is equidistant from

A

and

B

A = (1, 2, 3)

and

B = (2, 10, 1)

are two points. If the point

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and

P_{z} = 7

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P_{x} + P_{y} =

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A≡(1,2) and B≡(7,10) are two points. If P(x,y) is a point such that ∠APB=60∘ and area of ΔAPB is maximum, then which of the following is (are) TRUE ?

The correct options are B P lies on the perpendicular bisector of AB C P lies on the line 3x+4y=36Equation of perpendicular of AB isy−y1=−m(y−y1)y−6=−34(x−4)⇒3x+4y=36Also, AB=10,AL=5Therefore, AH= radius of circle=10√3=ALcosec60∘ and AL perpendicularly bisect the AB.

$A \equiv (1, 2)$ and $B \equiv (7, 10)$ are two points. If $P (x, y)$ is a point such that $∠ A P B = 60^{\circ}$ and area of $Δ A P B$ is maximum, then which of the following is (are) TRUE ?