RD Chapter 23 The Straight Lines Ex 23.6 Solutions
Question - 1 : - Find the equation to the straight line
(i) cutting off intercepts 3 and 2 from the axes.
(ii) cutting off intercepts -5 and 6 from the axes.
Answer - 1 : -
(i)┬аCutting off intercepts3 and 2 from the axes.
Given:
a = 3, b = 2
Let us find theequation┬аof line cutoff intercepts from the axes.
By using the formula,
The equation of theline is x/a + y/b = 1
x/3 + y/2 = 1
By taking LCM,
2x + 3y = 6
тИ┤ The equation of linecut off intercepts 3 and 2 from the axes is 2x + 3y = 6
(ii)┬аCutting off intercepts-5 and 6 from the axes.
Given:
a = -5, b = 6
Let us find theequation┬аof line cutoff intercepts from the axes.
By using the formula,
The equation of theline is x/a + y/b = 1
x/-5 + y/6 = 1
By taking LCM,
6x тАУ 5y = -30
тИ┤ The equation of linecut off intercepts 3 and 2 from the axes is 6x тАУ 5y = -30
Question - 2 : - Find the equation of the straight line which passes through (1, -2) andcuts off equal intercepts on the axes.
Answer - 2 : -
Given:
A line┬аpassingthrough (1, -2)
Let us assume, theequation of the line cutting equal intercepts at coordinates of length тАШaтАЩ is
By using the formula,
The equation of theline is x/a + y/b = 1
x/a + y/a = 1
x + y = a
The line x + y = apasses through (1, -2)
Hence, the pointsatisfies the equation.
1 -2 = a
a = -1
тИ┤ The equation of theline is x+ y = -1
Question - 3 : - Find the equation to the straight line which passes through the point (5,6) and has intercepts on the axes
(i) Equal in magnitude and both positive
(ii) Equal in magnitude but opposite in sign
Answer - 3 : -
(i)┬аEqual in magnitude andboth positive
Given:
a = b
Let us find theequation┬аof line cutoff intercepts from the axes.
By using the formula,
The equation of theline is x/a + y/b = 1
x/a + y/a = 1
x + y = a
The line passesthrough the point (5, 6)
Hence, the equationsatisfies the points.
5 + 6 = a
a = 11
тИ┤ The equation of theline is x + y = 11
(ii)┬аEqual in magnitude butopposite in sign
Given:
b = -a
Let us find theequation┬аof line cutoff intercepts from the axes.
By using the formula,
The equation of theline is x/a + y/b = 1
x/a + y/-a = 1
x тАУ y = a
The line passesthrough the point (5, 6)
Hence, the equationsatisfies the points.
5 тАУ 6 = a
a = -1
тИ┤ The equation of theline is x тАУ y = -1
Question - 4 : - For what values of a and b the intercepts cut off on the coordinate axesby the line ax + by + 8 = 0 are equal in length but opposite in signs to thosecut off by the line 2x тАУ 3y + 6 = 0 on the axes.
Answer - 4 : -
Given:
Intercepts cut off onthe coordinate axes by the line ax + by +8 = 0 тАжтАж (i)
And are equal inlength but opposite in sign to those cut off by the line
2x тАУ 3y +6 = 0 тАжтАж(ii)
We know that, theslope┬аof two lines is equal
The slope┬аof theline (i) is тАУa/b
The slope┬аof theline (ii) is┬а2/3
So let us equate,
-a/b = 2/3
a = -2b/3
The length of theperpendicular from the origin to the line (i) is
By using the formula,
The length of theperpendicular from the origin to the line (ii) is
By using the formula,
It is given that, d1┬а=d2
b = 4
So, a = -2b/3
= -8/3
тИ┤ The value of a is-8/3 and b is 4.
Question - 5 : - Find the equation to the straight line which cuts off equal positive intercepts on the axes and their product is 25.
Answer - 5 : -
Given:
a = b and ab = 25
Let us find theequation┬аof the line which cutoff intercepts on the axes.
тИ┤┬аa2┬а=25
a = 5 [consideringonly positive value of intercepts]
By using the formula,
The equation of theline with intercepts a and b is x/a + y/b = 1
x/5 + y/5 = 1
By taking LCM
x + y = 5
тИ┤ The equation of lineis x + y = 5