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The locus of the midpoint of the focal distance of a variable pointmoving on theparabola `y^2=4a x`is a parabola whoselatus rectum is half the latus rectum of the original parabolavertex is `(a/2,0)`directrix is y-axis.focus has coordinates (a, 0)A. latus rectum is half the latus rectum of the original parabolaB. vertex is (a/2,0)C. directrix is y-axisD. focus has coordinates (a,0) |
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Answer» Correct Answer - A::B::C::D 1,2,3,4 Any point on the parabola is `P(at^(2),2at)`. Therefore, the midpoint of S(a,0) and `P(at^(2),2at)` is `R((a+at^(2))/(2),at)-=(h,k)` `:.h=(a+at^(2))/(2),k=at` Eliminate t, i.e., `2x=a(1+(y^(2))/(a^(2)))=a+(y^(2))/(a)` `i.e.," " 2ax=a^(2)+y^(2)` `i.e," "y^(2)=2a(x-(a)/(2))` It is a parabola with vertex at (a/2,0) and latus rectum 2a. The directrix is `x-(a)/(2)=-(a)/(2)` `i.e," "x=0` The focus is `x-(a)/(2)=(a)/(2)` i.e, x=a i.e., (a,0) |
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