Chapter 6: Coordinate Geometry of Straight Lines (Set-1)
A line has slope 0. What is its direction?
A Vertical line
B Rising right
C Horizontal line
D Falling right
Slope 0 means no change in y when x changes, so the line is parallel to the x-axis. Hence it is a horizontal line everywhere.
For a vertical line x = 5, the slope is
A Undefined
B 0
C 1
D −1
In a vertical line, x does not change, so slope m = Δy/Δx has Δx = 0, which makes the slope undefined.
The slope of the x-axis is
A 1
B Undefined
C −1
D 0
The x-axis is a horizontal line. For any two points on it, y is the same, so Δy = 0 and slope becomes 0.
The slope of the y-axis is
A 0
B 1
C Undefined
D −1
The y-axis is vertical, so Δx = 0 between any two points on it. Therefore m = Δy/Δx is undefined.
Slope between (1,2) and (3,6) is
A 1
B 2
C 3
D 4
Slope m = (6−2)/(3−1) = 4/2 = 2. It shows y increases by 2 units for each 1 unit increase in x.
In y = mx + c, “c” represents
A x-intercept
B slope inverse
C y-intercept
D direction ratio
When x = 0 in y = mx + c, we get y = c. So c is the point where the line cuts the y-axis.
In y = mx + c, “m” represents
A slope
B y-intercept
C x-intercept
D distance
The coefficient of x is m, which equals the slope. It tells the rate of change of y with respect to x.
Equation with slope 3 through origin is
A y = x + 3
B x = 3y
C y = 3
D y = 3x
Through origin means c = 0 in y = mx + c. With slope m = 3, the equation becomes y = 3x.
A line parallel to x-axis has form
A x = a
B ax + by = 0
C y = b
D y = mx
A line parallel to x-axis has constant y-value for all x. So its equation is y = b, where b is the fixed y-coordinate.
A line parallel to y-axis has form
A x = a
B y = b
C y = mx + c
D ax + by = 0
A line parallel to y-axis keeps x constant for all y. So its equation is x = a, where a is the fixed x-coordinate.
Slope of line 2x + 3y + 6 = 0 is
A 2/3
B 3/2
C −2/3
D −3/2
In Ax + By + C = 0, slope m = −A/B. Here A = 2, B = 3, so m = −2/3.
Lines are parallel if their slopes are
A negative reciprocals
B equal
C product −1
D sum 0
Two distinct non-vertical lines are parallel when they have the same inclination, which happens when their slopes are equal.
Lines are perpendicular if m1·m2 equals
A −1
B 0
C 1
D 2
For two non-vertical lines, perpendicularity means angles differ by 90°. That condition is m1·m2 = −1.
If m = 2, a perpendicular slope is
A 2
B −2
C 1/2
D −1/2
Perpendicular slopes satisfy m·m⊥ = −1. So m⊥ = −1/2 when m = 2.
Slope of line joining (a,0) and (0,b) is
A a/b
B b/a
C −b/a
D −a/b
Slope m = (b−0)/(0−a) = b/(−a) = −b/a. This is also consistent with intercept form x/a + y/b = 1.
The two-point form uses
A two points
B one point, slope
C intercepts only
D distance only
Two-point form directly uses coordinates of two points on the line, allowing slope computation internally without needing slope separately.
Point-slope form of line is
A y = mx + c
B Ax + By + C = 0
C y − y1 = m(x − x1)
D x/a + y/b = 1
Point-slope form uses one known point (x1, y1) and slope m. It produces the line passing through that point with given slope.
Slope-intercept form is
A Ax + By + C = 0
B y = mx + c
C x = a
D y/b + x/a = 0
Slope-intercept form shows slope m and y-intercept c directly. It is the simplest form for graphing using intercept and slope.
Intercept form of a line is
A y = mx + c
B Ax + By = 0
C y − y1 = m(x − x1)
D x/a + y/b = 1
Intercept form uses x-intercept a and y-intercept b. It is useful when a line cuts both axes at known intercept values.
General form of a line is
A y = mx + c
B x/a + y/b = 1
C Ax + By + C = 0
D x = a
The general form represents any straight line (including vertical lines when B = 0). Coefficients A, B, C define the line uniquely up to scaling.
A line through (2,3) with slope 0 is
A y = 3
B x = 2
C y = x + 1
D x + y = 5
Slope 0 means horizontal line, so y remains constant. Passing through (2,3) fixes that constant as 3, giving y = 3.
A line through (2,3) perpendicular to x-axis is
A y = 3
B y = 2x
C x = 2
D 2x + y = 0
A line perpendicular to x-axis is vertical. Vertical lines have equation x = constant. Through (2,3), the constant is 2.
Slope of line x + 0y − 4 = 0 is
A Undefined
B 0
C 1
D −1
x − 4 = 0 represents x = 4, a vertical line. Vertical lines have undefined slope because Δx = 0.
If A = 0 in Ax + By + C = 0, line is
A vertical
B passing origin
C horizontal
D always slant
If A = 0, equation becomes By + C = 0 → y = −C/B, which is a constant y-value, hence a horizontal line.
Condition for collinearity of three points means
A same distance
B same midpoint
C same quadrant
D same straight line
Collinear points lie on one straight line. A common test is equal slopes between pairs or zero area of triangle formed by the points.
Angle between two lines with slopes m1, m2 uses
A tanθ = m1 + m2
B tanθ = (m1−m2)/(1+m1m2)
C tanθ = (m1m2)/(m1−m2)
D tanθ = 1+m1m2
The standard formula for angle between two non-vertical lines is tanθ = |(m1−m2)/(1+m1m2)|. It gives the acute angle by absolute value.
If two lines are parallel, their angle is
A 45°
B 90°
C 0°
D 180°
Parallel lines have the same direction and never meet. The angle between their directions is taken as 0° (or 180°), but usually the smaller is 0°.
If two lines are perpendicular, their angle is
A 90°
B 0°
C 30°
D 60°
Perpendicular lines intersect at a right angle. In slope terms, they satisfy m1·m2 = −1 (when both slopes are defined).
If m1m2 = 1 and m1 ≠ m2, lines are
A parallel
B perpendicular
C acute angle
D equal lines
m1m2 = 1 does not mean perpendicular (needs −1). With different slopes, they intersect and typically make an acute angle depending on values.
If m1 = m2, tanθ becomes
A 1
B −1
C Undefined
D 0
With m1 = m2, numerator (m1−m2) becomes 0, so tanθ = 0 and θ = 0°, meaning the lines are parallel or coincident.
Normal form includes “p” as
A slope value
B perpendicular distance
C y-intercept
D x-intercept
In normal form x cosα + y sinα = p, p is the perpendicular distance from origin to the line along its normal direction.
Direction ratios of a line with slope m are
A (m,1)
B (−m,1)
C (1,m)
D (1,−m)
A line with slope m rises m units for 1 unit run. So a direction vector can be (1, m), giving direction ratios 1 and m.
A normal vector to Ax + By + C = 0 is
A (A,B)
B (B,−A)
C (−B,A)
D (1,m)
In Ax + By + C = 0, coefficients (A, B) are perpendicular to the line, so they form a normal vector.
Distance from (x1,y1) to ax+by+c=0 is
A |ax1+by1+c|
B √(a²+b²)/|ax1+by1+c|
C |ax1+by1+c|/√(a²+b²)
D |ax1+by1|/c
Perpendicular distance formula divides the absolute value of the line expression at the point by √(a²+b²), which normalizes using the line’s normal vector length.
Distance from (0,0) to 3x+4y−10=0 is
A 1
B 2
C 10/√25
D 10/5
Use distance = |c|/√(a²+b²) for origin. Here |−10|/√(3²+4²) = 10/5 = 2, so option 10/5 matches.
If ax1+by1+c = 0, point is
A above line
B below line
C on line
D farthest point
A point satisfies the line equation exactly when substituted values make it zero. That means the point lies on the line.
For parallel lines, coefficients satisfy
A a1 = b2
B a1/b1 = a2/b2
C a1a2 + b1b2 = 0
D c1 = c2
Two lines a1x+b1y+c1=0 and a2x+b2y+c2=0 are parallel when their normal vectors are proportional, i.e., a1/b1 = a2/b2.
Distance between ax+by+c1=0 and ax+by+c2=0 is
A |c1−c2|
B |c1+c2|
C |c1−c2|/√(a²+b²)
D √(a²+b²)/|c1−c2|
For parallel lines with same a and b, shortest distance equals difference in constants divided by √(a²+b²), which is length of normal vector.
Distance between x−2=0 and x+3=0 is
A 5
B 1
C 3
D 6
These are vertical parallel lines x=2 and x=−3. Distance between them is |2−(−3)| = 5 units along the x-direction.
Distance between y=4 and y=−1 is
A 3
B 4
C 6
D 5
These are horizontal parallel lines. Distance is the absolute difference in y-values: |4−(−1)| = 5.
Line through (1,2) parallel to 3x−2y+5=0 is
A 3x+2y+1=0
B 2x−3y+1=0
C 3x−2y+1=0
D 3x−2y−1=0
Parallel lines keep same x and y coefficients. Put (1,2) in 3x−2y+c=0 → 3−4+c=0 gives c=1.
Line through (1,2) perpendicular to 3x−2y+5=0 has form
A 2x+3y+c=0
B 3x−2y+c=0
C 2x−3y+c=0
D 3x+2y+c=0
For Ax+By+C=0, slope is −A/B. A perpendicular line swaps coefficients to (B,−A) type; here perpendicular family is 2x+3y+c=0.
Intersection point of x=2 and y=3 is
A (3,2)
B (2,3)
C (2,0)
D (0,3)
x=2 fixes x-coordinate as 2, and y=3 fixes y-coordinate as 3. The unique point satisfying both is (2,3).
If two lines intersect, they are not
A straight
B linear
C parallel
D coordinate
Parallel lines do not meet. If two lines intersect at one point, they must have different directions, so they cannot be parallel.
Midpoint of (2,4) and (6,8) is
A (3,6)
B (4,8)
C (2,6)
D (4,6)
Midpoint is average of coordinates: ((2+6)/2, (4+8)/2) = (4,6). It lies exactly halfway along the segment.
A point divides segment internally in ratio m:n means
A outside segment
B same point
C inside segment
D no division
Internal division means the point lies between the two endpoints. Ratio m:n describes how the segment is split into two parts along the line joining endpoints.
Area of triangle with collinear points is
A zero
B 1 square unit
C maximum
D infinite
If points are collinear, they lie on one line and cannot form a triangle with height. Therefore, the area becomes zero.
Centroid of triangle is intersection of
A angle bisectors
B altitudes
C medians
D perpendicular bisectors
The centroid is where all three medians meet. A median joins a vertex to the midpoint of the opposite side, and all medians intersect at one point.
Equation of perpendicular bisector uses
A slope only
B y-intercept only
C area only
D endpoints only
A perpendicular bisector passes through the midpoint of a segment and is perpendicular to the segment. So it is determined from the two endpoints of the segment.
If line passes through origin, its equation can be
A y=mx+c, c≠0
B x/a+y/b=1 always
C Ax+By+C=0, C=0
D x=k, k≠0 always
Passing through origin means (0,0) satisfies the equation, so C must be 0 in Ax+By+C=0. Then the line is Ax+By=0 through the origin.