RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.1 (Updated 2021-22) RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.1: “A straight line is a curve such that every point on the line segment joining any two points lies on it is termed as a straight line”. In this chapter, we will discuss concepts related to straight lines, with examples for better understanding. RD Sharma Class 11 Maths Solutions contains all the solutions to the math problems given in the textbook. The subject matter experts have accurately prepared and solved the questions in each section. Students can consult and download RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.1 in PDF from the link given below and start offline practice.

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Straight Lines Ex 23.1 Q1 Straight Lines Ex 23.1 Q2 Straight Lines Ex 23.1 Q3(i) Straight Lines Ex 23.1 Q3(ii) Straight Lines Ex 23.1 Q3(iii) Straight Lines Ex 23.1 Q3(iv) Straight Lines Ex 23.1 Q4 Straight Lines Ex 23.1 Q5(i) Straight Lines Ex 23.1 Q5(ii) Straight Lines Ex 23.1 Q6 Straight Lines Ex 23.1 Q7 Straight Lines Ex 23.1 Q8 Straight Lines Ex 23.1 Q9 Straight Lines Ex 23.1 Q10 Straight Lines Ex 23.1 Q11 Straight Lines Ex 23.1 Q12 Straight Lines Ex 23.1 Q13 Straight Lines Ex 23.1 Q14 Straight Lines Ex 23.1 Q15 Straight Lines Ex 23.1 Q16 Straight Lines Ex 23.1 Q17 Straight Lines Ex 23.1 Q18 Straight Lines Ex 23.1 Q19 Straight Lines Ex 23.1 Q20 Straight Lines Ex 23.1 Q21 RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.1: Important Topics From The Chapter

First Degree Equation

Every first degree equation like ax+by+c=0ax+by+c=0 would be the equation of a straight line.

Slope of a line

• Slope (m) of a non-vertical line passing through the points (x1 ,y1 )(x1 , y1 ) and (x2 , y2)(x2 , y2) is given by is given by m =y1−y2x1−x2= m =y1−y2x1−x2=  x1≠x2x1≠x2.
• If a line makes an angle á with the positive direction of x-axis, then the slope of the line is given by m =tanα, α≠90om =tanα, α≠90o
• Slope of horizontal line is zero and slope of vertical line is undefined.
• An acute angle (say θ) between lines L1 and L2L1 and L2with slopes m1 and m2m1 and m2 is given by tanθ=∣∣m2−m11+m1m2∣∣tan⁡θ=|m2−m11+m1m2|, 1+m1m2≠01+m1m2≠0
• Two lines are parallel if and only if their slopes are equal i.e., m1=m2m1=m2
• Two lines are perpendicularif and only if product of their slopes is –1, i.e., mm2=−1m1.m2=−1
• Three points A, B and C are collinear, if and only if slope of AB = slope of BC.
• Equation of the horizontal line having distance a from the x-axis is eithery = a or y = – a.
• Equation of the vertical line having distance b from the y-axis is eitherx = b or x = – b.
• The point (x, y) lies on the line with slope m and through the fixed point (xo,y0 ),(xo, y0 ), if and only if its coordinates satisfy the equation.

Various forms of equations of a line:

• Two points form: Equation of the line passing through the points (x1,y1)(x1, y1) and ((x2, y2)(x2, y2) is given by y−y1=y2−y1x2−x1(x−x1)y−y1=y2−y1x2−x1(x−x1)
• Slope-Intercept form: The point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y=mx +cy=mx +c.
• If a line with slope m makes x-intercept d. Then equation of the line is y=m(x -d)y=m(x -d).
• Intercept form: Equation of a line making intercepts a and b on the x-and y-axis, respectively, is xa+yb=1xa+yb=1.
• Normal form: The equation of the line having normal distance from origin p and angle between normal and the positive x−axis ωx−axis ωis given by  x cosω +ysin ω=p x cosω +ysin ω=p
• General Equation of a Line: Any equation of the form Ax + By + C = 0, with A and B are not zero, simultaneously, is called the general linear equation or general equation of a line.
• Working Rule for reducing general form into the normal form:

(i) Shift constant ‘C’ to the R.H.S. and get Ax+By=−CAx+By=−C
(ii) If the R.H.S. is not positive, then make it positive by multiplying the whole equation by -1.
(iii) Divide both sides of equation by A2+B2−−−−−−−√A2+B2.

The equation so obtained is in the normal form.

• Parametric Equation (Symmetric Form): x−x1cosθ=y−y1sinθ=rx−x1cos⁡θ=y−y1sin⁡θ=r
• Equation of a line through origin: y=mxy=mxor y=xtanθy=xtan⁡θ.
• The perpendicular distance (d) of a line Ax + By+ C = 0 from a point (x1, y1)  (x1, y1) is given by d=|Ax1+By1+C|A2+B2√d=|Ax1+By1+C|A2+B2
• Distance between the parallel lines Ax + By + C1Ax + By + C1= 0 and Ax + By + C2Ax + By + C2= 0, is given by d=|C1−C2|A2+B2√d=|C1−C2|A2+B2

Concurrent Lines
Three of more straight lines are said to be concurrent if they pass through a common point i.e., they meet at a point. Thus, if three lines are concurrent the point of intersection of two lines lies on the third line.

Condition of concurrency of three lines:
a1(b2c3−b3c2)+b1(c2a3−c3a2)+c1(a2b3−a3b2)=0a1(b2c3−b3c2)+b1(c2a3−c3a2)+c1(a2b3−a3b2)=0

EQUATIONS OF FAMILY OF LINES THROUGH THE INTERSECTION OF TWO LINES
A1x+B1y+C1+k(A2x+B2y+C2)=0A1x+B1y+C1+k(A2x+B2y+C2)=0
where kk is a constant and also called parameter.
This equation is of first degree of xx and yy, therefore, it represents a family of lines.

DISTANCE BETWEEN TWO PARALLEL LINES
Working Rule to find the distance between two parallel lines:
(i) Find the co-ordinates of any point on one of ht egiven line, preferably by putting x=0x=0 and y=0y=0.
(ii) The perpendicular distance of this point from the other line is the required distance between the lines.

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