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The CORRECT choice is (d) y=-5

To explain: EQUATION of x-axis is y=0. Horizontal line is parallel to x-axis and below it by 5 UNITS so, equation of line is y=-5.

Correct choice is (b) False

To explain: We KNOW, SLOPE of LINE joining TWO points (x1, y1) and (x2, y2) is GIVEN by(y2-y1)/(x2-x1).

Slope of line AB = (5-2)/(3-1) = 3/2

Slope of line BC = (8-5)/(7-3) = 3/4

Since slope of AB is not equal to BC so, points are not collinear.

The CORRECT answer is (c) M1 * m2 = -1

To explain: If the two lines are perpendicular then if one line form ANGLE α with positive x-axis then the other line form angle 90° + α.

If m1 = TAN α then m2 will be tan (90°+ α) = – cot α = -1/tan α

=> m1 * m2 = – 1.

Right answer is (d) Not DEFINED

Best EXPLANATION: If a line is parallel to y-axis then angle formed by it with x-axis is zero. So, its inclination is 90°. Hence SLOPE = tan 90° which is not defined.

Right answer is (b) 27 sq. units

To EXPLAIN I would say: We know, area of TRIANGLE joining vertices (x1, y1), (x2, y2) and (x3, y3) is (1/2)* determinant \(\begin{pmatrix}-4 & -4 &1 \\-3& 2 & 1\\ 3& -16 & 1\end{pmatrix}\) is \(\frac{1}{2}\){(-4)(2+16) – (-4)(-3-3) + (1)(48-6)} = \(\frac{1}{2}\)|(-72)+(-24)+42| = 27 square units.

Right choice is (a) True

To EXPLAIN: Let the inclination of the two lines be α and β. SINCE they are parallel so, α = β.

=>TAN α = tan β. Hence their SLOPES are equal.

The CORRECT option is (a) 0°

To explain I would SAY: If a line is parallel to x-axis then angle FORMED by it with x-axis is ZERO. So, its inclination is zero.

The CORRECT choice is (C) y=5

Best EXPLANATION: Equation of x-axis is y=0. HORIZONTAL line is parallel to x-axis and above it by 5 units so, equation of line is y=5.

The CORRECT choice is (a) 2

For explanation: We know, slope of line joining two points (x1, Y1) and (X2, y2) is GIVEN by (y2-y1)/(x2-x1).

So, slope of line joining (0, 0) and (3, 6) is (6-0)/(3-0) = 6/3 = 2.

The CORRECT ANSWER is (c) tanα

For explanation I would say: Slope of a line is given by tanα if inclination of line is α. Slope is DENOTED by tangent of the inclination ANGLE.

The correct CHOICE is (C) √3

To elaborate: Slope of a line is given by tanα if INCLINATION of line is α. If inclination is 60° the slope is TAN 60° = √3.

The correct choice is (d) 13 UNITS

The EXPLANATION is: We KNOW, distance between two points (x1, Y1) and (x2, y2) is\(\SQRT{(x_1-x_2)^2+(y_1-y_2)^2}\).

So, distance between (5, 12) from origin (0, 0) is \(\sqrt{(5-0)^2+(12-0)^2} = \sqrt{(5)^2+(12)^2}\) = 13 units.

Correct choice is (d) -2

Easy explanation: GIVEN equation is y=5x+10. X-intercept MEANS value of x when y is zero

0=5x+10 => x=-2

Right option is (b) 2:1

The best explanation: The coordinates of a POINT dividing the line segment joining (x1, Y1) and (X2, y2) internally in the ratio m: n is \((\FRAC{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})\).

Let the ratio be k: 1.So, the coordinates of a point dividing the line segment joining (1, 2) and (4, 5) internally in the ratio k: 1 is \((\frac{k*4+1*1}{k+1},\frac{k*5+1*2}{k+1})\)

=> \((\frac{k*4+1*1}{k+1},\frac{k*5+1*2}{k+1})\) is same as (3, 4).

=> (4k+1)/(k+1) = 3

=> 4k+1 = 3k+3

=> k = 2

So, ratio is 2:1.

The correct CHOICE is (c) y=4

For explanation: LET general equation of LINE be y=m*x + c.

Since line is PARALLEL to x-axis so, m=0.

=>y=c=>y=4 by substituting the point (3, 4).

The correct choice is (a) True

Easiest explanation: AREA of TRIANGLE FORMED by THREE points is zero then the three points must be collinear i.e. they must lie on the same line.

Right CHOICE is (d) y = 2x-2

The best I can explain: We know, equation of LINE joining two POINTS (x1, y1) and (x2, y2) is given by

\(\FRAC{y-y1}{y2-y1} = \frac{x-x1}{x2-x1}\)

So, equation will be \(\frac{y-4}{8-4} = \frac{x-3}{5-3}\)

=>\(\frac{y-4}{4} = \frac{x-3}{2}\)

=>y-4 = 2x-6

=>y = 2x-2

The correct option is (a) (3, 4)

EXPLANATION: The coordinates of a point DIVIDING the line segment JOINING (x1, y1) internally in the ratio m: N is \((\FRAC{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})\).

So, the coordinates of a point dividing the line segment joining (1, 2) and (4, 5) internally in the ratio 2:1 is \((\frac{2*4+1*1}{2+1}, \frac{2*5+1*2}{2+1})\) = (3, 4).

Right answer is (b) \(\frac{3}{\SQRT{5}}\)

For EXPLANATION: Distance of point (x1, y1) from LINE ax +by +c=0 is \(|\frac{ax1+by1+c}{\sqrt{a^2+b^2}}|\).

So, distance of point (2, 4) from line 2x+y-5=0 is \(|\frac{2*2+4*1-5}{\sqrt{2^2+1^2}}| = \frac{3}{\sqrt{5}}\).

Correct choice is (a) √2

To elaborate: Given EQUATION is x + y=2. REDUCING the above equation to NORMAL form

(x + y)/√2 =√2.

x cos⁡45° + y SIN 45° = √2

Perpendicular distance from ORIGIN is √2.

Right choice is (a) x + y = 4√2

Best EXPLANATION: If p is perpendicular DISTANCE of line from ORIGIN and ω be the angle FORMED by normal with positive x-axis then equation of line is x cos⁡ω + y sin⁡ω = p.

So, equation of given line will be x cos⁡45° + y sin⁡45° = 4

=>(x + y)/√2 = 4

=>x + y = 4√2.

Right OPTION is (c) 5 units

The explanation: We know, distance between two points (x1, y1) and (x2, y2) is\(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\).

So, distance between (1, 3) and (5, 6) is \(\sqrt{(1-5)^2+(3-6)^2}=\sqrt{(4)^2+(3)^2}\) = 5 units.

The correct option is (a) y = 4x-8

The BEST EXPLANATION: Let general equation of line be y=m*x + c.

Given m=4 and VALUE of x when y=0 is 2.

C= – m*2 = -4*2 = -8.

=>y=4x-8.

Right choice is (a) 3

To explain I would SAY: Comparing the above EQUATION with general equation y=m*X + C,

m=3 which is the slope of line.

The correct option is (c) X+2y-10 = 0

To explain: LINE 2x – y=4 has SLOPE 2. Line PERPENDICULAR to given line has slope -1/2.

Equation is (y-4) = (-1/2) (x-2)

2y-8 = -x + 2 => x+2y -10 = 0.

Right OPTION is (d) 14

The explanation is: We know, slope of line JOINING two POINTS (x1, y1) and (x2, y2) is given by(y2-y1)/(x2-x1).

Lines are parallel means slope is equal.

=>(x-4)/(11-3) = (7-2)/(5-1) => x-4 = 5*8/4 = 10 => x=14.

Right CHOICE is (d) 90°

Explanation: If a LINE is parallel to y-axis then ANGLE FORMED by it with x-axis is 90°. So, its inclination is 90°.

The CORRECT answer is (a) 104°F

The explanation is: LET GENERAL equation be F=m*C +k

-40 = -40m + k

and 32 = 0 + k

=>-40 = -40m + 32 => m=72/40 = 18/10

F=18/10 * 40 + 32 = 72+32 = 104.

The correct choice is (b) 3

The explanation: We know, SLOPE of line joining two points (x1, Y1) and (X2, y2) is GIVEN by \(\frac{y_2-y_1}{x_2-x_1}\).

So, slope of line joining (1, 2) and (4, 11) is \(\frac{11-2}{4-1} = \frac{9}{3}\) = 3.

Right option is (d) y=3x-1

To EXPLAIN I would say: LET general EQUATION of line be y=m*x + c.

Given m=3 => y=3x+c

Substituting the point (1, 2) in above equation we get 2=3*1 + c => c = -1

So, equation of line will be y = 3x-1.

Right choice is (a) x=5

Easiest EXPLANATION: EQUATION of y-axis is x=0. Vertical line is PARALLEL to y-axis and to the right by 5 UNITS so, equation of line is x=5.

Right option is (c) -16

Easy EXPLANATION: We know, slope of line JOINING two points (x1, y1) and (x2, y2) is given by(y2-y1)/(x2-x1).

Lines are perpendicular means m1*m2 = -1

=> \((\FRAC{X-4}{11-3})(\frac{6-2}{7-1})\) = -1

=> (x-4)(4) = (-1)(8)(6)

=> x-4 = -12 => x= -16.

Correct answer is (B) x+2y-4=0

For explanation I WOULD say: If x-intercept of a line is a and y-intercept of line is b so, equation of line is \(\FRAC{x}{a} + \frac{y}{b}\) = 1.

Equation of line is \(\frac{x}{4} + \frac{y}{2}\) = 1 => x+2y-4=0.

The correct answer is (a) -3/2

Best explanation: If LINES with slopes m1 and M2 are PERPENDICULAR then m1 * m2 = – 1.

If m1 = 2/3 then m2 = -1 / (2/3) = -3/2.

Right ANSWER is (C) 90°

Explanation: If the two LINES are perpendicular then if one line form ANGLE α with positive x-axis then the other line form angle 90°+ α.

The CORRECT answer is (b) (3, 5)

For EXPLANATION I would say: We know, midpoint of (x1, y1) and (x2, y2) is \((\FRAC{x1+x2}{ 2}, \frac{y1+y2}{2})\).

So, midpoint of (1, 2) and (5, 8) is ((1+5)/2, (2+8)/2) is (3, 5).

The correct CHOICE is (a) 4x+y-13=0

For EXPLANATION I would say: Line 4x+y=2 has SLOPE -4. Line parallel to it has slope -4 and pass through (2, 5) so equation will be y-5 = (-4) (x-2) => 4x+y-13=0

Correct option is (a) 5x-4y+3 = 0

The best I can explain: MIDPOINT of QR line is \((\frac{3+7}{2}, \frac{5+9}{2})\) = (5, 7).

Equation of line joining (1, 2) and (5, 7) is \(\frac{y-2}{7-2} = \frac{x-1}{5-1}\)

=>\(\frac{y-2}{5} = \frac{x-1}{4}\) => 4y-8 = 5x-5 => 5x-4y+3 =0.

Correct choice is (C) obtuse angle

To ELABORATE: If inclination is α SLOPE is given by TAN α. Given that slope of line is negative which MEANS tan α is negative. We know, tan α is negative in 2^nd quadrant i.e. α should be obtuse angle.

Correct CHOICE is (b) False

Best explanation: If two LINES are parallel then they form same ANGLE with POSITIVE direction of x-axis in anticlockwise direction i.e. their inclinations are equal.