Connect the curve 3

Calculus Level 4

In this figure, the complete black circle is a circle of equation ( x − 9 ) 2 + ( y − 7 ) 2 = 2 { \left( x-9 \right) }^{ 2 }+{ \left( y-7 \right) }^{ 2 }=2

The red line below it is of the equation x=8, with range restrictions 0 < y ≤ 4 0<y\le4 .

If AB touches both the circle and the red straight line and is the shortest of its kind, then its length can be written as A ( B − C ) \sqrt { A } \left( \sqrt { B } -C \right)

If A, B, C are integers with A and B being square free, find A+B+C.

See the full image from here


The answer is 8.

Archit Boobna by Archit Boobna , 20, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Rohit Sachdeva
Apr 18, 2015

For BA to be smallest A should lie on circle as well as line connecting B & C (centre of circle)

B(8,4) & C(9,7)

Let A divide BC in ration m:1

Then A(9m+8/m+1,7m+4/m+1)

Since A lies on circle, on simplifying we get

10/(m+1)²=2 or m= 5 − 1 \sqrt{5}-1

Now, BC= 10 \sqrt{10}

AB=m/(m+1) x BC = 2 ( 5 − 1 ) \sqrt{2}(\sqrt{5}-1)

So A+B+C=8

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...