Distance Formula

Q. Consider a parabola $y=\frac{x^2}{4}$ and the point $F(0,1)$. Let $A_1(x_1,y_1),A_2(x_2,y_2),A_3(x_3,y_3),...,A_k(x_k,y_k)$ are $\text{'}n\text{'}$ points on parabola such as $x_k>0$ and $\angle OF{A}_k=\frac{k\pi }{2n},(k=1,2,3,...,n)$. Then the value of $\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^{n}F{A}_k$ is

1. $\frac{2}{\pi }$
2. $\frac{4}{\pi }$
3. $\frac{8}{\pi }$
4. $\frac{16}{\pi }$

Q. The shortest distance between the point $(\frac{3}{2},0)$ and the curve $y=\sqrt{x},(x>0),$ is :

1. $\frac{\sqrt{5}}{2}$
2. $\frac{3}{2}$
3. $\frac{5}{4}$
4. $\frac{\sqrt{3}}{2}$

Q. The length of the latus rectum of a conic is $5$. Its focus is $(-1,1)$ and its directrix is $3x-4y+2=0$, then the conic is

1. a parabola
2. an ellipse
3. a hyperbola
4. a circle

Q.

Find the locus of the point P if $A{P}^2-B{P}^2=18$, where A ≡ (1, 2, –3) and B ≡ (3, –2, 1)

1. 2x + 3y + 4 = 9

2. 2x – 4y + 4z = 9

3. 2x + 4y + 4z = 9

4. 2x – 3y + 4z = 9

Q.

What is the distance between the point A(0, 7, 10) and C(–4, 9, 6) in space?

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Q. A point $P$ in space is such that the sum of square of its distances from $x-$axis, $y-$axis and $z-$axis is $9\text{units}$ more than the sum of square of its distances from $xy-$plane, $yz-$plane and $zx-$plane. Then the distance( in $\text{units}$) of point $P$ from origin is equal to

Q. The point equidistant from the points (a, 0, 0), (0, b, 0), (0, 0, c) and (0, 0, 0) is

1. $(\frac{a}{3},\frac{b}{3},\frac{c}{3})$
2. $(a,b,c)$
3. $(\frac{a}{2},\frac{b}{2},\frac{c}{2})$
4. $(4a,4b,4c)$

Q. Points (1, 1, 1), (–2, 4, 1), (–1, 5, 5) and (2, 2, 5) are the vertices of a

1. Parallelogram
2. Trapezium
3. Rectangle
4. Square

Q.

Find the locus of the point P if $A{P}^2-B{P}^2=18$, where A ≡ (1, 2, –3) and B ≡ (3, –2, 1)

1. 2x + 3y + 4 = 9

2. 2x – 4y + 4z = 9

3. 2x + 4y + 4z = 9

4. 2x – 3y + 4z = 9

Q.

The length of perpendicular drawn from the point (5, 4, -1) to the line $\overrightarrow{r}=\hat{i}+\lambda (2\hat{i}+9\hat{j}+5\hat{k})$ is

1. none of these

2. $\sqrt{\frac{2001}{112}}$

3. $\sqrt{\frac{2309}{112}}$

4. $\sqrt{\frac{2109}{110}}$