First slide

Functions (XII)

Question

If $f : R \to N \cup \{0\},$ where f (area of triangle joining points $P (5, 0), Q (8, 4) and R (x, y)$ such that angle $P R Q$ is a right angle) = number of triangles, then which of the following is true?

Moderate

A

$f (5) = 4$
B

$f (7) = 0$
C

$f (6.25) = 2$
D

$f (4.5) = 4$

Solution

$\begin{array}{l} Hence ∠ P R Q = π / 2 and points P, Q, R lie on the circle with P Q as diameter . \\ Also, P Q = 5 . \end{array}$
Now, the maximum area of the triangle is
$Δ_{\max} = \frac{1}{2} (5) (\frac{5}{2}) = 6.25$
For area = 5, we have four symmetrical positions of point
$R (shown as R_{1}, R_{2}, R_{3}, R_{4}) .$
For area = 6.25, we have exactly two points.
For area: 7 , no such points exist.

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