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Perpendicular Distance of a Point from a Plane

Let Z 1, Z 2,...

Question

Let $Z_{1}, Z_{2}, Z_{3}$ be three points $A, B$ and $P$ respectively in the argand plane. Let $P$ moves in the plane such that $| Z - Z_{1} | + | Z - Z_{2} | = K .$ Let $Z_{1} = - Z_{2} = i$

If $K = 6$ and maximum area of the triangle $A B P$ is $\sqrt{8}$ sq. units, then possible number of positions of $P$ is

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Solution

The correct option is A
2

$| Z - Z_{1} | + | Z - Z_{2} | = K$ is the condition of an ellipse.

It is a standing ellipse with foci $(0, 1)$ and $(0, - 1)$

$\Rightarrow b e = 1$ [Comparing it with standard equation of ellipse : $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with eccentricity, $e$ ]

$\Rightarrow b \times \sqrt{1 - \frac{a^{2}}{b^{2}}} = 1$

$\Rightarrow \sqrt{b^{2} - a^{2}} = 1$

$\Rightarrow b^{2} - a^{2} = 1$

Now, $b = 3 ∵ K = 6$

$\Rightarrow a = 2 \sqrt{2}$

$A B = 2$

Maximum area of the triangle $A B P = 2 \sqrt{2}$ sq. units $= \frac{1}{2} \times A B \times a$

Thus, possible number of positions for $P$ is 2 which are $(2 \sqrt{2}, 0)$ or $(- 2 \sqrt{2}, 0)$

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Let Z1, Z2, Z3 be three points A, B and P respectively in the argand plane. Let P moves in the plane such that |Z−Z1|+|Z−Z2|=K. Let Z1=–Z2=i If K=6 and maximum area of the triangle ABP is √8 sq. units, then possible number of positions of P is