We will learn how to solve different types of problems on straight lines.
1. Find the angle which the straight line perpendicular to the straight line √3x + y = 1, makes with the positive direction of the xaxis.
Solution:
The given equation of the straight line √3x + y = 1
Covert the above equation into slopeintercept form we get,
y =  √3x + 1…………………… (i)
Let us assume that the given straight line (i) makes an angle θ with the positive direction of the xaxis.
Then, the slope of the straight line (i) will be tan θ
Hence, we must have,
tan =  √3 [Since, the slope of the straight line y =  √3x + 1 is  √3]
⇒ tan θ =  tan 60° = tan (180°  60°) = tan 120°
⇒ tan θ = 120°
Since the straight line (i) makes an angle 120° with the positive direction of the xaxis, hence a straight line perpendicular to the line (i) will make an angle 120°  90° = 30° with the positive direction of the xaxis.
2. Prove that P (4, 3), Q (6, 4), R (5, 6) and S (3, 5) are the angular points of a square.
Solution:
We have,
PQ = \(\sqrt{(6  4)^{2} + (4  3)^{2}}\) = √5
QR = \(\sqrt{(6  4)^{2} + (5  4)^{2}}\) = √5
RS = \(\sqrt{(5  6)^{2} + (3  5)^{2}}\) = √5 and
SP = \(\sqrt{(5  3)^{2} + (3  4)^{2}}\) = √5
Therefore, PQ = QR = RS = SP.
Now, m\(_{1}\) = Slope of PQ = \(\frac{4  3}{6  4}\) = ½
m\(_{2}\) = Slope of QR = \(\frac{6  4}{5  6}\) = 2 and
m\(_{3}\) = Slope of RS = \(\frac{5  6}{3  5}\) = ½
Clearly, m\(_{1}\) ∙ m\(_{2}\) = ½ ∙ (2) = 1 and m\(_{1}\) = m\(_{3}\).
This shows that PQ is perpendicular to QR and PQ is parallel to RS.
Thus, PQ = QR = RS = SP, PQ ⊥ QR and PQ is parallel to RS.
Thence, PQRS is a square.
3. A straight line passes through the point ( 1, 4) and makes an angle 60° with the positive direction of the xaxis. Find the equation of the straight line.
Solution:
The required line makes an angle 60° with the positive direction of the axis of x.
Therefore, the slope of the required line = m = tan 60° = √3. Again, the required line passes through the point ( 1, 4).
Therefore, the equation of the required straight line is
y  4 = √3(x + 1), [Using the pointslope form, y  y\(_{1}\) = m (x  x\(_{1}\))].
4. Find the equation of the straight line which passes through the point (5, 6) and has intercepts on the axes equal in magnitude but opposite in sign. Find also the coordinates of the point on the line at which the ordinate is double the abscissa.
Solution:
Let us assume that, the equation of the required straight line be
\(\frac{x}{a}\) + \(\frac{y }{b}\) = 1 ………………. (i)
According to the question, b =  a; hence, equation (i) reduces to
\(\frac{x}{a}\) + \(\frac{y }{a}\) = 1
⇒ x  y = a ………………. (ii)
Again, the line (ii) passes through the point (5, 6). Therefore,
5  6 = a
⇒ a =  1
Therefore, the equation of the required straight line is,
x y = 1
⇒ x y + 1 = 0………………. (iii)
Now, we are to find the coordinates of that point on the line (iii) for which the ordinate is double the abscissa.
Let the coordinates of the required point be (α, β). Then the point (α, β) will satisfy the equation (iii).
Therefore, α  2α + 1 = 0
⇒ α = 1.
Therefore, the coordinates of the required point are (1, 2).
● The Straight Line
11 and 12 Grade Math
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