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If `(h ,k)`is a point on the axis of the parabola `2(x-1)^2+2(y-1)^2=(x+y+2)^2`from where three distinct normals can be drawn, then prove that `h > 2.`A. `hgt2`B. `hlt4`C. `hgt8`D. `hlt8` |
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Answer» Correct Answer - A We have, `2(x-1)^(2)+(y-1)^(2)=(x+y+2)^(2)` `rArr" "sqrt((x-1)^(2)+(y-1)^(2))=|(x+y+2)/(sqrt(1+1))|^(2)` Clearly, it represents a parabola having its focus at (1, 1) and directrix x + 2 = 0. The equation of the axis is `y-1(x-1) i.e. y = x` Semi-latusrectum = Length of `bot` from (1, 1) on the directrix `rArr" Semi-laturectum "=|(1+1+2)/(sqrt(1+1))|=2sqrt2`. The coordinates of the vertex are (0, 0). So, the equation of the axis in parametrix form if `(x-0)/("cos "pi//4)=(y-0)/("sin "pi//4)" ...(i)"` We know that three distinct normals can be drawn from a point (h, 0) on the axis of the parabola `y^(2)=4ax` if `hgt2a("= semi-latusretum")`. The coordinates of a point on the axis (i) at a distance `2 sqrt2` from the vertex are given by `x/("cos "pi//4)=y/("sin "pi//4)=2sqrt2rArrx=2, y=2`. Hence, three normals can be drawn from (h, k), if `hgt2`. |
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