Saved Bookmarks
| 1. |
The tangent PT and the normal PN to the parabola `y^2=4ax` at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose:A. `"vertex is "(2a//3, 0)`B. `"Directri is x = 0"`C. `"Latusrectum is "(2a)/3`D. `"Focus is "(-a, 0)` |
|
Answer» Correct Answer - A Let `P(at^(2), 2at)` be any point on the parabola `y^(2)=4ax`. The equations of tangent and normal at P are `ty=x+at^(2)" ...(i)"` `"and, "y+tx=2at+at^(2)" ...(ii)"` These two meet x-axis i.e. axis of the parabola at `T(-at^(2), 0)` and `N(2a+at^(2), 0)` respectively. Let (h, k) be the coordinates of the centroid of `DeltaPTN`. Then, `h=(2a+at^(2))/3" and "k=(2at)/3` `rArr" "3h=2a+a((3k)/(2a))^(2)" [On eliminating t]"` `rArr" "12ah=8a^(2)+9k^(2)` Hence, the locus of (h, k) is `12ax=8a^(2)+9y^(2)` `rArr" "y^(2)=(4a)/3(x-(2a)/3)` Clearly, it represents a parabola whose vertex is at (2a/3, 0). Latusrectum `=(4a)/3`, focus `((2a)/3+a/3,0)-(a, 0)` and directrix `x=a//3`. |
|