Saved Bookmarks
| 1. |
Show that the locus of a point that divides a chord of slope 2 of the parabola `y^2=4x` internally in the ratio `1:2` is parabola. Find the vertex of this parabola. |
|
Answer» Correct Answer - Vertex `-=((2)/(9),(8)/(9))` Slope of the chord joining point `P(t_(1))andQ(t_(2))` on the parabola `y^(2)=4x` is `(2)/(t_(2)+t_(1))=2` (given) `:." "t_(2)+t_(1)=1` (1) Let point R(h,k) divide PQ in the ratio 1 : 2. `So," "h=(t_(2)^(2)+2t_(1)^(2))/(1+2)andk=(2t_(2)+2(2t_(1)))/(1+2)` `rArr" "3h=t_(2)^(2)+2t_(1)^(2)` (2) `and" "t_(2)+2t_(1)=(3k)//2` (3) From (1) and (3), we get `t_(1)=(3)/(2)k-1andt_(2)=2-(3)/(2)k` Putting these values in (2), we get `(2-(3)/(2)k)^(2)+2((3)/(2)k-1)^(2)=3h` Hence, locus of the point R is `(y-(8)/(9))^(2)=(4)/(9)(x-(2)/(9))`, which is parabola having vertex at `((2)/(9),(8)/(9))`. |
|