A straight line passing through A ( - 2 , 1 ) makes an angle of 30 ° with O X ↔ in the positive direction. Find the points on the straight line whose distance from A is 4 units.

Let, x 1 , y 1 = A ( - 2 , 1 ) , r = 4 and α = 30 ° Then, cos α = cos 30 ° = 3 2 , sin α = sin 30 ° = 1 2 Co-ordinates of point at a distance r from x 1 , y 1 on the given line are x 1 ± r cos α , y 1 ± r sin α . Co-ordinates of the required points are - 2 + 4 · 3 2 , 1 + 4 · 1 2 = - 2 + 2 3 , 3 and - 2 - 4 · 3 2 , 1 - 4 · 1 2 = - 2 - 2 3 , - 1 .

A straight line whose inclination with the positive direction of the X - axis, measured in the anti-clockwise sense is π 3 makes positive intercept on the Y - axis. If the straight line is at a distance of 4 from the origin, find its equation.

Let α = π 3 , p = 4 Slope m = tan α = tan 60 ° = 3 Equation of the line in the slope-intercept form is y = 3 x + c ⇒ 3 x - y + c = 0 Distance from the origin is = 4 ⇒ | 0 - 0 + c | 3 + 1 = 4 ⇒ | c | = 4 × 2 = 8 ⇒ c = ± 8 Given c > 0 ∴ c = 8 The required equation is 3 x - y + 8 = 0 .

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IMPORTANT

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If the product of the intercepts made by the straight line $x \tan α + y \sec α = 1 (0 \leq α < \frac{π}{2})$ on the coordinate axes is equal to $\sin α,$ find $α$ .

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Important Questions on The Straight Line

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11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 3 - The Straight Line>Exercise 3(b)>Q II

If the sum of the reciprocals of the intercepts made by a variable straight line on the axes of coordinates is a constant, then prove that the line always passes through a fixed point.

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11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 3 - The Straight Line>Exercise 3(b)>Q II

Line

L

has intercepts

a

and

b

on the axes of coordinates. When the axes are rotated through a given angle, keeping the origin fixed, the same line

L

has intercepts

p

and

q

on the transformed axes. Prove that

\frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{p^{2}} + \frac{1}{q^{2}}

HARD

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 3 - The Straight Line>Exercise 3(b)>Q II

Transform the equation

\frac{x}{a} + \frac{y}{b} = 1

into the normal form when

a > 0

and

b > 0

. If the perpendicular distance of the straight line from the origin is

p

, deduce that

\frac{1}{p^{2}} = \frac{1}{a^{2}} + \frac{1}{b^{2}}

EASY

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IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 3 - The Straight Line>Exercise 3(b)>Q III

A straight line passing through

A (- 2, 1)

makes an angle of

30 °

with

\overset{\leftrightarrow}{O X}

in the positive direction. Find the points on the straight line whose distance from

A

4

units.

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 3 - The Straight Line>Exercise 3(b)>Q III

Find the points on the line

3 x - 4 y - 1 = 0

which are at a distance of

5

units from the point

(3, 2)

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 3 - The Straight Line>Exercise 3(b)>Q III

A straight line whose inclination with the positive direction of the

X -

axis, measured in the anti-clockwise sense is

\frac{π}{3}

makes positive intercept on the

Y -

axis. If the straight line is at a distance of

4

from the origin, find its equation.

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 3 - The Straight Line>Exercise 3(b)>Q III

A straight line

L

is drawn through the point

A (2, 1)

such that its point of intersection with the straight line

x + y = 9

is at a distance of

3 \sqrt{2}

from

A

. Find the angle which the line

L

makes with the positive direction of the

X -

axis.

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 3 - The Straight Line>Exercise 3(b)>Q III

A straight line

L

with negative slope passes through the point

(8, 2)

and cuts positive coordinate axes at the points

P

and

Q

. Find the minimum value of

O P + O Q

L

varies, where

O

is the origin.

If the product of the intercepts made by the straight line xtanα+ysecα=10≤α<π2 on the coordinate axes is equal to sinα, find α.

Important Questions on The Straight Line

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If the product of the intercepts made by the straight line $x \tan α + y \sec α = 1 (0 \leq α < \frac{π}{2})$ on the coordinate axes is equal to $\sin α,$ find $α$ .