Question

If `H` is the othrocenter of an acute angled triangle ABC whose circumcircle is `x^2+y^2=16 ,` then circumdiameter of the triangle HBC is 1 (b) 2 (c) 4 (d) 8

Answer 1

With the given details, we can create a diagram.
Please refer to video for the diagram.
Here, circumradius of `Delta ABC` = `sqrt16 = 4`
`:. a = 2*4sinA =>a = 8sinA->(1)`
Let `R_1` is the circumradius of the `Delta HBC`.
Then,
`a = 2R_1sin/_BHC`
`=2R_1sin(B+C)`
`=2R_1sin(pi-A)`
`:. a=2R_1sinA->(2)`
From (1) and (2),
`2R_1sinA = 8sinA`
`=>2R_1 = 8`
`:.` Circumdiameter of `Delta HBC = 2R_1 = 8.`