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Question Number 216139 by mr W last updated on 28/Jan/25

Commented by mr W last updated on 28/Jan/25

$f i n d {(A P + P Q)}_{m i n} = ?$
Commented by Ghisom last updated on 28/Jan/25

$seems to be 6$
Commented by mr W last updated on 28/Jan/25

$i t h i n k t o o .$
	Answered by mahdipoor last updated on 28/Jan/25

	$i f P = (x, {a x}^{2}) i s f i x e d \Rightarrow$ $Q P i s m i n w h e n P, Q, B i n o n e l i n e$ $\Rightarrow Q P = P B - Q B = P B - r$ $s = A P + P Q =$ $\sqrt{x^{2} + {({a x}^{2} - 2)}^{2}} + \sqrt{{(x - 3)}^{2} + {({a x}^{2} - 6)}^{2}} - 2$ $\frac{d s}{d x} = 0 \Rightarrow x = 3 \Rightarrow s_{m i n} = 8$
	Answered by mr W last updated on 29/Jan/25

	Commented by mr W last updated on 29/Jan/25

	$A = f o c u s o f p a r a b o l a$ $B Q = r = 2$ $Missing \left or extra \right$ ${(A P + P Q + Q B)}_{m i n} = {(A P + P B)}_{m i n} = {(C P + P B)}_{m i n} = 6 + 2 = 8$ ${(A P + P Q)}_{m i n} = 8 - r = 6 ✓$
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