Perpendicular Distance of a Point from a Line

Q.

Given $A=si{n}^2\theta +co{s}^4\theta$ then for all real values of $\theta$

1. $1\le A\le 2$

2. $\frac{3}{4}\le A\le 1$

3. $\frac{13}{16}\le A\le 1$

4. $\frac{3}{4}\le A\le \frac{13}{16}$

Q.

Find a relation between $x$ and $y$ such that the point $\left(x,y\right)$ is equidistant from the point $\left(3,6\right)$ and $\left(-3,4\right)$.

Q.

Distance between the lines $5x+3y-7=0$ and $15x+9y+14=0$ is

1. $\frac{1}{3\sqrt{34}}$

2. $\frac{35}{3\sqrt{34}}$

3. $\frac{35}{2\sqrt{34}}$

4. $\frac{35}{\sqrt{34}}$

Q.

If for some $\mathrm{\alpha }\in \mathrm{R}$, the lines ${\mathrm{L}}_1:\frac{\left(\mathrm{x}+1\right)}{2}=\frac{\left(\mathrm{y}-2\right)}{-1}=\frac{(\mathrm{z}-1)}{1}$ and ${\mathrm{L}}_2:\frac{(\mathrm{x}+2)}{\mathrm{\alpha }}=\frac{\left(\mathrm{y}+1)\right)}{5-\mathrm{\alpha }}=\frac{(\mathrm{z}+1)}{1}$are coplanar, then the line ${\mathrm{L}}_2$ passes through the point

1. $(2,-10,-2)$

2. $(10,-2,-2)$

3. $(10,2,2)$

4. $(-2,10,2)$

Q. The distance of the point $(-2,4,-5)$ from the line $\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}$ is

1. $\sqrt{\frac{37}{10}}$
2. $\frac{37}{\sqrt{10}}$
3. $\frac{37}{10}$
4. $\frac{\sqrt{37}}{10}$

Q.

The values of $\lambda$, so that the line $3x-4y=\lambda$ touches $x^2+y^2-4x-8y-5=0$ are

1. $-35,15$

2. $3,-5$

3. $35,-15$

4. $-3,5$

5. $20,5$

Q. The length of the perpendicular from the point $(2,-1,4)$ on the straight line, $\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}$ is :

1. less than $2$
2. greater than $2$ but less than $3$
3. greater than $4$
4. greater than $3$ but less than $4$

Q. If the equation of the line passing through $M(1,1,1)$ and intersecting at right angle to the line of intersection of the planes $x+2y-4z=0$ and $2x-y+2z=0$ is $\frac{x-1}{a}=\frac{y-1}{b}=\frac{z-1}{c},$ then $a:b:c$ equals

1. $5:-1:2$
2. $-5:1:2$
3. $5:1:-2$
4. $5:1:2$

Q. Let $\sqrt{3}\hat{i}+\hat{j},\hat{i}+\sqrt{3}\hat{j}$ and $\beta \hat{i}+(1-\beta )\hat{j}$ respectively be the position vectors of the points $A,B$ and $C$ with respect to the origin $O$. If the distance of $C$ from the bisector of the acute angle between $OA$ and $OB$ is $\frac{3}{\sqrt{2}}$, then the sum of all possible values of $\beta$ is :

1. $1$
2. $2$
3. $3$
4. $4$

Q.

If the length of the perpendicular drawn from the origin to the line whose intercepts on the axes are $a$ and $b$ be $p$, then

1. $a^2+b^2=p^2$

2. $a^2+b^2=\frac{1}{p^2}$

3. $\frac{1}{a^2}+\frac{1}{b^2}=\frac{2}{p^2}$

4. $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{p^2}$

Q.

If the point P on the curve, $4x^2+5y^2=20$ is farthest from the point Q$(0,-4)$, then PQ² is equal to:

1. $48$

2. $29$

3. $21$

4. $36$

Q. The distance of the point having position vector $-\hat{i}+2\hat{j}+6\hat{k}$ from the straight line passing through the point $(2,3,-4)$ and parallel to the vector, $6\hat{i}+3\hat{j}-4\hat{k}$ is :

1. $6$
2. $7$
3. $2\sqrt{13}$
4. $4\sqrt{3}$

Q. If $p_1$ and $p_2$ are the lengths of the perpendiculars from the origin to the straight lines $x\sec \theta +y\text{cosec}\theta =a$ and $x\cos \theta -y\sin \theta =a\cos 2\theta$ respectively, then the value of $4p_1^2+p_2^2$ is

1. $4a^2$
2. $2a^2$
3. $a^2$
4. $\frac{a^2}{2}$

Q.

The equation of the perpendicular bisector of the line segment joining $A(2,3)$ and $B(6,-5)$ is

1. $x–y=–1$

2. $x–2y=3$

3. $x+y=3$

4. $x-2y=6$

Q.

Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to the line x - 2y = 1.

Q.

The values of $\lambda$, for which the equation$x^2-y^2-x+\lambda y-2=0$ represents a pair of straight lines, is

1. $-3,1$

2. $-1,1$

3. $3,-3$

4. $3,3$

Q. Equation(s) of the common tangent to the parabola $y^2=24x$ and the circle $x^2+y^2=18$ is/are

1. $2x+y+3=0$
2. $x-y+6=0$
3. $x+y+6=0$
4. $x-2y+8=0$

Q.

The three lines $\mathrm{lx}+\mathrm{my}+\mathrm{n}=0,\mathrm{mx}+\mathrm{ny}+\mathrm{l}=0,\mathrm{nx}+\mathrm{ly}+\mathrm{m}=0$ are concurrent if

1. $\mathrm{l}=\mathrm{m}+\mathrm{n}$

2. $\mathrm{m}=\mathrm{l}+\mathrm{n}$

3. $\mathrm{n}=\mathrm{l}+\mathrm{m}$

4. $\mathrm{l}+\mathrm{m}+\mathrm{n}=0$

Q.

Let $L_1$ be a tangent to the parabola$y^{{}_2}=4(x+1)$ and $L_2$ be a tangent to the parabola $y^2=8(x+2)$ such that $L_1\text{and}{L}_2$ intersect at right angles. Then $L_1\text{and}{L}_2$ meet on the straight line:

1. $x+2y=0$

2. $x+2=0$

3. $2x+1=0$

4. $x+3=0$

Q.

The value of $\lambda$, for which the equation $x^2-y^2-x-\lambda y-2=0$ represents a pair of straight lines, are

1. $3,-3$

2. $-3,1$

3. $3,1$

4. $-1,1$

Q. If one end of a focal chord of the parabola, $y^2=16x$ is at $(1,4)$, then the length of this focal chord is:

1. $25$
2. $24$
3. $22$
4. $20$

Q. The perpendicular bisector of the line segment joining $P(1,4)$ and $Q(k,3)$ has y-intercept $-4$. Then a possible value of $k$ is

1. 2
2. 1
3. -2
4. -4

Q. The shortest distance between the line $y=x$ and the curve $y^2=x-2$ is:

1. $\frac{11}{4\sqrt{2}}$
2. $\frac{7}{4\sqrt{2}}$
3. $\frac{7}{8}$
4. $2$

Q. The equation of perpendicular bisectors of sides $AB,BC$ of $\Delta ABC$ are $x-y-5=0$ and $x+2y=0$ respectively. If $A\equiv (1,-2),C\equiv (\alpha ,\beta )$, then $\alpha +\beta$ is equal to

Q.

A point equidistant from the lines $4x+3y+10=0$, $5x-12y+26=0$ & $7x+24y-50=0$ is

1. $(1,-1)$

2. $(1,1)$

3. $(0,0)$

4. $(0,1)$

Q. The locus of the point of intersection of tangents to the parabola $y^2=4ax$ which includes an angle $\alpha$ is

1. $y^2-4ax=(x+a{)}^2{\cot }^2\alpha$
2. $y^2-4ax=(x-a{)}^2{\tan }^2\alpha$
3. $y^2-4ax=(x+a{)}^2{\tan }^2\alpha$
4. $y^2-4ax=(x-a{)}^2{\cot }^2\alpha$

Q. The equation of circle which passes through focus of parabola $x^2=4y$ and touches it at $(6,9)$ is

1. $x^2+y^2+24x-27y+26=0$
2. $x^2+y^2+48x-12y+11=0$
3. $x^2+y^2+18x-22y+21=0$
4. $x^2+y^2+18x-28y+27=0$

Q.

If $(\alpha ,\beta )$ is the point on $y^2=6x$, that is closest to $\left(3,\frac{3}{2}\right)$ then find $2(\alpha +\beta )$

1. $6$

2. $9$

3. $7$

4. $5$

Q.

The normal at point $\left(1,1\right)$ of the curve $y^2=x^3$ is parallel to the line

1. $3x-y-2=0$

2. $2x+3y-7=0$

3. $2x-3y+1=0$

4. $2y+3x-1=0$

Q. If $y+b=m_1(x+a),y+b=m_2(x+a)$ are two tangents to the parabola $y^2=4ax,$ then

1. $m_1\cdot m_2=-1$
2. $m_1\cdot m_2=1$
3. $m_1+m_2=0$
4. $m_1-m_2=0$