We will learn how to find the equation of a straight line in normal form.
The equation of the straight line upon which the length of the perpendicular from the origin is p and this perpendicular makes an angle α with xaxis is x cos α + y sin α = p
If the line length of the perpendicular draw from the origin upon a line and the angle that the perpendicular makes with the positive direction of xaxis be given then to find the equation of the line.
Suppose the line AB intersects the xaxis at A and the yaxis at B. Now from the origin O draw OD perpendicular to AB.
The length of the perpendicular OD from the origin = p and ∠XOD = α, (0 ≤ α ≤ 2π).
Now we have to find the equation of the straight line AB.
Now, from the rightangled ∆ODA we get,
\(\frac{OD}{OA}\) = cos α
⇒ \(\frac{p}{OA}\) = cos α
⇒ OA = \(\frac{p}{cos α}\)
Again, from the rightangled ∆ODB we get,
∠OBD = \(\frac{π}{2}\)  ∠BOD = ∠DOX = α
Therefore, \(\frac{OD}{OB}\) = sin α
or, \(\frac{p}{OB}\) = sin α
or, OB = \(\frac{p}{sin α}\)
Since the intercepts of the line AB on xaxis and yaxis are OA and OB respectively, hence the required
\(\frac{x}{OA}\) + \(\frac{y}{OB}\) = 1
⇒ \(\frac{x}{\frac{p}{cos α}}\) + \(\frac{y}{\frac{p}{sin α}}\) = 1
⇒ \(\frac{x cos α}{p}\) + \(\frac{y sin α}{p}\) = 1
⇒ x cos α + y sin α = p, which is the required form.
Solved examples to find the equation of a straight line in normal form:
Find the equation of the straight line which is at a of distance 7 units from the origin and the perpendicular from the origin to the line makes an angle 45° with the positive direction of xaxis.
Solution:
We know that the equation of the straight line upon which the length of the perpendicular from the origin is p and this perpendicular makes an angle α with xaxis is x cos α + y sin α = p.
Here p = 7 and α = 45°
Therefore, the equation of the straight line in normal form is
x cos 45° + y sin 45° = 7
⇒ x ∙ \(\frac{1}{√2}\) + y ∙ \(\frac{1}{√2}\) = 7
⇒ \(\frac{x}{√2}\) + \(\frac{y}{√2}\) = 7
⇒ x + y = 7√2, which is the required equation.
Note:
(i) The equation of a, straight line in the form of x cos α + y sin α = p is called its normal form.
(ii) In equation x cos α + y sin α = p, the value of p is always positive and 0 ≤ α≤ 360°.
11 and 12 Grade Math
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