Equation of a Line Perpendicular to a Given Line

Q. If 2x − 3y = 7 and (a + b)x − (a + b − 3)y = 4a + b represent coincident lines, then a and b satisfy the equation

(a) a + 5b = 0
(b) 5a + b = 0
(c) a − 5b = 0
(d) 5a − b = 0

Q. Solve the following simultaneous equations graphically.
(1) 2x + 3y = 12 ; x – y = 1
(2) x – 3y = 1 ; 3x – 2y + 4 = 0
(3) 5x – 6y + 30 = 0 ; 5x + 4y – 20 = 0
(4) 3x – y – 2 = 0 ; 2x + y = 8
(5) 3x + y = 10 ; x – y = 2

Q. ABCD is a rectangle formed by joining the points A (−1, −1), B(−1 4) C (5 4) and D (5, −1). P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

Q. The coordinates of the circumcentre of the triangle formed by the points O (0, 0), A (a, 0 and B (0, b) are

(a) (a, b)

(b) $\left(\frac{a}{2},\frac{b}{2}\right)$

(c) $\left(\frac{b}{2},\frac{a}{2}\right)$

(d) (b, a)

Q. Solve the following systems of equations:

$$\begin{gathered} \frac{5}{x+1}-\frac{2}{y-1}=\frac{1}{2} \\ \frac{10}{x+1}+\frac{2}{y-1}=\frac{5}{2} \end{gathered}$$

where x ≠ −1 and y ≠ 1

Q. Draw the graph of the equations x = 3 , x = 5 and 2x − y − 4 = 0 . Also, find the area of the quadrilateral formed by the lines and the x-axis.

Q.

The equation of a line is 3x + 4y – 7 = 0. Find :

(i) the slope of the line

(ii) the equation of a line perpendicular to the given line and passing through the intersection of the lines x – y + 2 = 0 and 3x + y -10 = 0.

Q. Find the point on x-axis which is equi-distance from (2, -5)and (-2, 9)

Q.

If the coordinates of the point of intersection of less than ogive and more than ogive is (12.5, 20) then the value of median is

1. 20

2. 12.5

3. 6.5

4. 12

Q.

If the line passing through $(4,3)$ and $(2,k)$ is perpendicular to $y=2x+3$, then $k=?$

1. $-1$

2. $1$

3. $-4$

4. $4$

Q. If points A (5, p) B (1, 5), C (2, 1) and D (6, 2) form a square ABCD, then p =

(a) 7

(b) 3

(c) 6

(d) 8

Q. Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, −1) , (1, 3) and (x, 8) respectively.

Q.

Find the equation of the perpendicular bisector of the line segment joining the points $(3,4)$ and $(-1,2)$.

1. $2y–x=5$

2. $y–x=5$

3. $3y–2x=5$

4. $y+2x=5$

Q. Find the value of p for which the lines $2x+3y-7=0$ and $4y-px-12=0$ are perpendicular to each other.

1. -4

2. 6

3. -6

4. 4

Q. If A (2, 2), B (−4, −4) and C (5, −8) are the vertices of a triangle, than the length of the median through vertex C is

(a) $\sqrt{65}$

(b) $\sqrt{117}$

(c) $\sqrt{85}$

(d) $\sqrt{113}$

Q.

Find the value of k if the lines given by 3x+2ky=2 ; 2x+5y=12 are parallel

Q. Find the equation of a line passing through the point of intersection of 4x+2y=7, with the y axis and parallel to y-7x+5=0

Q.

If A$(4,9)$, B$(2,3)$, and C$(6,5)$ are the vertices of ∆ABC, then the length of the median through C is
(a) $5$ units

(b) $\sqrt{10}$ units

(c) $\sqrt{25}$ units

(d) $10$ units

Q.

A car travels at a speed of 60 km/hr. Write the distance – time formula and draw the graph of it. Also, find the distance travelled by the car in 5 hours.

Q.

Find the equation of the line whose -intercept is 8 and -intercept is -12.

1. 

2. 

3. 

4. 

Q.

A line cuts the x - axis at A(7, 0) and the y - axis at B(0, -5). A variable line PQ is drawn perpendicular to AB. Cutting the x - axis at P and the y - axis at Q.
If AQ and BP intersect at R, the locus of R is

1. $x^2+y^2+7x-5y=0$

2. $x^2+y^2-7x+5y=0$

3. 5x - 7y = 35

4. none of these

Q.

Prove that any line parallel to the parallel sides of a trapezium divides the non-parallel sides proportionally.

Q. If the lines $(p+1)x+y=3$ and $3y-(p-1)x=4$ are perpendicular to each other, find $p$.

1. 1

2. 2

3. -1

4. -2

Q.

A plane makes intercepts $a,b,c$ at $A,B,C$ on the coordinates axes, respectively. If the centroid of $∆ABC$ is at $(3,2,1)$, then the equation of the plane is

1. $x+2y+3z=9$

2. $2x–3y–6z=18$

3. $2x+3y+6z=18$

4. $2x+y+6z=18$

5. $2x+3y+6z=9$

Q.

The line 4x - 3y + 12 = 0 meets x-axis at A. Write the co-ordinates of A.

Determine the equation of the line through A and perpendicular to 4x - 3y + 12 = 0.

Q.

Find the point where 2x-3y = -9 intersects the y axis. [1 MARK]

Q.

The vertices of a triangle are A(–2, 8), B(1, 2) and C(7, –1).

i) Find the equation of the median AD.

ii) Find the equation of the altitude BE.

1. (4, 1/2), x-y+1=0

2. (2, 1/4), 2x-y+1=0

3. (4, 1/2), x-2y+1=0

4. (1, 4/2), x-y+1=0

Q.

The point P is the foot of perpendicular from A (-5, 7) to the line 2x - 3y + 18 = 0.

Determine :

(i) the equation of the line AP

(ii) the co-ordinates of P

Q.

A, B and C have co-ordinates (0, 3), (4, 4) and (8, 0) respectively. Find the equation of the line through A and perpendicular to BC.

Q.

The slope of the side BC of a rectangle ABCD is $\frac{2}{3}$. Find :

(i) the slope of the side AB,

(ii) the slope of the side AD.