Two Point Form of a Line

Q. Suppose that the points $(h,k),(1,2)$ and $(-3,4)$ lie on the line $L_1$. If a line $L_2$ passing through the points $(h,k)$ and $(4,3)$ is perpendicular to $L_1$, then $\frac{k}{h}$ equals :

1. $0$
2. $3$
3. $\frac{1}{3}$
4. $-\frac{1}{7}$

Q. If in a parallelogram $ABDC$, the coordinates of $A,B$ and $C$ are respectively $(1,2),(3,4)$ and $(2,5)$, then the equation of the diagonal $AD$ is :

1. $5x-3y+1=0$
2. $5x+3y-11=0$
3. $3x-5y+7=0$
4. $3x+5y-13=0$

Q.

The inclination of the straight line passing through the point $\left(-3,6\right)$ and the midpoint of the line joining the $\left(4,-5\right)$ and $\left(-2,9\right)$ is

1. $\frac{\mathrm{\pi }}{4}$

2. $\frac{\mathrm{\pi }}{6}$

3. $\frac{\mathrm{\pi }}{3}$

4. $\frac{3\mathrm{\pi }}{4}$

Q. The locus represented by $xy+yz=0$ is

1. a pair of perpendicular lines
2. a pair of parallel lines
3. a pair of parallel planes
4. a pair of perpendicular planes

Q. The equation of the median through the vertex $A$ of triangle $ABC$ whose vertices are $A(2,5),B(-4,9)$ and $C(-2,-1)$ is

1. $x-5y+23=0$
2. $7x+4y-34=0$
3. $8x-y+11=0$
4. $x+3y-17=0$

Q.

Find the equation of the straight line passing through the points$(-1,1)$ and$(2,-4)$.

Q. If each of the points $(x_1,4),(-2,y_1)$ lies on the line joining the points $(2,-1)$ and $(5,-3)$, then the point $P(x_1,y_1)$ lies on the line

1. $2x+6y+1=0$
2. $6(x+y)-25=0$
3. $6(x+y)+25=0$
4. $2x+3y-6=0$

Q.

State whether the two lines in each of the following are parallel, perpendicular or neither:

(i) Through (5, 6) and (2, 3); through (9, -2) and (6, -5)

(ii) Through (9, 5) and (-1, 1); through (3, -5) and (8, -3)

(iii) Through (6, 3) and (1, 1); through (-2, 5) and (2, -5)

(iv) Through (3, 15) and (16, 6); through (-5, 3) and (8, 2).

Q.

The equation of a line passing through $(-2,-4)$ and perpendicular to the line $3x-y+5=0$ is:

1. $3y+x-8=0$

2. $3x+y+6=0$

3. $x+3y+14=0$

4. None of these

Q.

The equation of a straight line which passes through the point $(-3,5)$ such that the portion of it between the axes is divided by the point in the ratio $5:3$ (reckoning from the x-axis) will be

1. $x+y-2=0$

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

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

4. $x-y+8=0$

Q. The equation of the internal bisector of the angle A of a triangle ABC whose vertices are A(4, 3), B(0, 0) and C(2, 3) is .

1. x-3y+5=0
2. x+3y-5=0
3. 3x-y-9=0
4. 3x+y-9=0

Q.

Find the equation of the side BC of the triangle ABC whose vertices are A (-1, -2), B (0, 1) and C (2, 0) respectively. Also, find the equation of the median through A (-1, -2).

Q.

Find the equations to the straight lines which go through the origin and trisect the portion of the straight line 3x + y = 12 which is intercepted between the axes of coordinates.

Q. In a triangle $ABC\text{if}A\equiv (1,2)$ and internal angle bisectors through B and C are $y=x$ and $y=-2x,$ then the inradius r of the $△ABC$ is

Q.

In a hyperbola the distance between the foci is three times the distance between the directories then its eccentricity is

1. 5/2

2. 5/4

3. 3/2

4. 

Q.

Find the equation of the bisector of angle A of the triangle whose vertices are A (4, 3), B (0, 0) and C (2, 3).

Q.

Find the equations of the diagonals of the square formed by the lines x = 0, y = 0, x = 1 and y = 1.

Q.

The length L (in centimeters) of a copper rod is a linear function of its Celsius temperature C. In an experiment, if L = 124.942 when C = 20 and L = 125.134 when C = 110, express L interms of C.

Q. The equation of straight line which is equidistant from the points $A(2,–2)$, $B(6,1)$ and $C(–3,4)$ can be

1. $2x+6y-5=0$
2. $12x+10y-43=0$
3. $6x-8y-11=0$
4. $6x-8y+11=0$

Q.

Find the equation of the striaght lines passing through the following pair of points:

(i) (0, 0) and (2, - 2) (ii) (a, b) and ( a + c sin $\alpha$, b + c cos $\alpha$)

(iii) (0, - a) and (b, 0) (iv) (a, b) and (a + b, a - b) (v) $(a{t}_1,a/t_1)and(a{t}_2,a/t_2)$ (vi) $(acos\alpha ,asin\alpha )and(acos\beta ,asin\beta )$

Q.

The equation of the bisector of the angle between two lines $3x-4y+12=0$ and $12x-5y+7=0,$ which contain the point $(-1,4)$ is

Q.

In what ratio is the line joining the points (2, 3) and (4, -5) divided by the line passing through the points (6, 8) and (-3, -2).

Q.

Elena and her husband Marc both drive to work. Elenas car has a current mileage (total distance driven) of $9,000$ and she drives $22,000$ $\mathrm{miles}$ more each year. Marcs car has a current mileage of $22,000$ and he drives $22,000$ $\mathrm{miles}$ more each year. Will the mileage for the two cars ever be equal? Explain.

1. No; The equations have different slopes, so the lines do not intersect.

2. Yes; The equations have different $y-$intercepts, so the lines intersect.

3. No; The equations have different slopes, so the lines intersect.

4. No; The equations have equal slopes but different $y-$intercepts, so the lines do not intersect.

Q.

The vertices of a quadrilateral are A (-2, 6), B (1, 2), C (10, 4) and D (7, 8). Find the equations of its diagonals.

Q.

A vertex of square is $\left(3,4\right)$ and diagonal $x+2y=1$, then the second diagonal which passes through given vertex will be

1. $2x–y+2=0$

2. $x+2y=11$

3. $2x–y=2$

4. none of these

Q.

Find the equation to the straight line which bisects the distance between the points (a, b), (a', b') and also bisects the distance between the points (- a, b) and a', - b).

Q.

Find the equations to the diagonals of the rectangle the equations of whose sides are x = a, x = a', y = b and y = b'.

Q.

The number of points on the line $x+y=4$, which are a unit distance apart from the line $2x+2y=5$, is

1. $0$

2. $1$

3. $2$

4. $\infty$

Q.

The point of intersection of the lines represented by the equation $2x^2+3y^2+7xy+8x+14y+8=0$ is

1. (0, 2)

2. (1, 2)

3. (-2, 0)

4. (-2, 1)

Q.

In the triangle ABC with vertices A(2, 3), B(4, -1) and C(1, 2) find the equation and the length of the altitude from the vertex A.