Let A 1-2 and B 7-10 are two points- If P x - y is a point such that the - APB is 60- and the area of the APB is maximum- then which of the following is-are true-A- P lies on any line perpendicular to ABB- P lies on the perpendicular bisector of ABC- P lies on the straight line 3 x-4 y-36D- P lies on the circle passing through the points 1-2 and 7-10 and having a radius of 10 units

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Byju's Answer

Standard XII

Mathematics

Equation of Circle with (h,k) as Center

Let A 1,2 and...

Question

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.

P lies on any line perpendicular to AB

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P lies on the perpendicular bisector of AB

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P lies on the straight line 3x + 4y = 36

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P lies on the circle passing through the points (1, 2) and (7, 10) and having a radius of 10 units

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Solution

The correct options are
B
P lies on the perpendicular bisector of AB

C
P lies on the straight line 3x + 4y = 36

For the area to be maximum, P should lie on the perpendicular bisector of the side AB.

Mid-point of AB= $(\frac{7 + 1}{2}, \frac{10 + 2}{2}) = (4, 6)$

Slope (AB)= $\frac{8}{6} = \frac{4}{3}$

$∴$ Slope of $⊥$ bisector = $- \frac{3}{4}$

Thus equation of line on which P lies is

$y - 6 = - \frac{3}{4}$ or $3 x + 4 y = 36$

Since, the angle subtended is constant, AB would a chord with P being the trace of the circle.

For max area, $△ A B P$ would be equilateral.

Length of $A B = 10$

$∴$ Radius of the circle $= \frac{2}{3} \cdot \frac{\sqrt{3}}{2} \cdot 10 = \frac{10 \sqrt{3}}{3}$

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

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.