Question

An insect walks in x-y plane from A(12,1) in a straight line to reach a point P on x axis.From point P
it walks to another point B(9,4) again in a straight line. Let x be the length of shortest path it
travels, find sum of digits of 100(x^2+ x^4)

Answer 1

Answer:

11

Step-by-step explanation:

An insect walks in x-y plane from A(12,1) in a straight line to reach a point P on x axis

=> P = ( P , 0)  as its on x axis

Distance PA = √( (P - 12)² + (0-1)²) =  √ ((P - 12)² + 1)

PB = √( (P - 9)² + (0-4)²)  = √ ((P - 9)² + 16)

x = PA + PB  = √ ((P - 12)² + 1) + √ ((P - 9)² + 16)

dx/dP = (1/2)2(P - 12)/√ ((P - 12)² + 1)   +  (1/2)2(P-9)/√ ((P - 9)² + 16)

dx/dP = 0

=> (1/2)2(P - 12)/√ ((P - 12)² + 1)  = - (1/2)2(P-9)/√ ((P - 9)² + 16)

=> (P - 12)√ ((P - 9)² + 16) = -(P-9)√ ((P - 12)² + 1)

Squaring both sides

(P-12)²(P - 9)² + 16)  = (P -9)² ((P - 12)² + 1) )

=> (P-12)²(P - 9)² + 16(P-12)² = (P -9)²(P - 12)²   +  (P -9)²

=> 16(P-12)²  = (P -9)²

Taking square root both sides

=> 4(P - 12) = ± (P-9)

=> P = 13  or P = 11.4

P = 11.4 will give shortest Distance

x = √ ((P - 12)² + 1) + √ ((P - 9)² + 16)

= √(0.6² + 1) + √(2.4)² + 16

= √1.36 + √21.76

= √1.36(1  + 4)

= 5√1.36

=> x² = 34

100 (x² + x⁴)

= 100 ( 34  + (34)²)

= 3400 ( 1 + 34)

= 35 * 3400

= 119000

Sum of Digits = 1 + 1 + 9  = 11

= 100 ( 34 + 24