If h , r is the foot of the perpendicular from x 1 , y 1 to l x + m y + n = 0 , prove that x 1 - h l = y 1 - r m = l x 1 + m y 1 + n l 2 + m 2 .

Let P h , r is the foot of the perpendicular from A x 1 , y 1 to l x + m y + n = 0 . So, P h , r lies on the line l x + m y + n = 0 ⇒ l h + m r + n = 0 ⇒ l h + m r = - n Slope of line A P = r - y 1 h - x 1 Slope of line l x + m y + n = 0 is - l m . Since the line A P is perpendicular to line l x + m y + n = 0 . ⇒ r - y 1 h - x 1 - l m = - 1 ⇒ y 1 - r x 1 - h = m l ⇒ x 1 - h l = y 1 - r m = l x 1 - h + m y 1 - r l 2 + m 2 = l x 1 + m y 1 - l h - m r l 2 + m 2 = l x 1 + m y 1 + n l 2 + m 2 Hence, x 1 - h l = y 1 - r m = l x 1 + m y 1 + n l 2 + m 2 .

E M B I B E

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 2 - Straight Lines>2.8>Q 25

HARD

JEE Main

IMPORTANT

Earn 100

If the image of the point $(2, 1)$ with respect to the line mirror be $(5, 2)$ , find the equation of the mirror.

Important Questions on Straight Lines

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 2 - Straight Lines>2.8>Q 26

If

(h, r)

is the foot of the perpendicular from

(x_{1}, y_{1})

to

l x + m y + n = 0

, prove that

\frac{x_{1} - h}{l} = \frac{y_{1} - r}{m} = \frac{l x_{1} + m y_{1} + n}{l^{2} + m^{2}}

.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 2 - Straight Lines>2.8>Q 27

Find the equation of the straight line passing through the point

(2, - 6)

and the point of intersection of the lines

5 x - 2 y + 14 = 0

and

2 y = 8 - 7 x

.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 2 - Straight Lines>2.8>Q 28

Find the equation of the straight line which passes through the point

(1, 1)

and the point of intersection of the lines

3 x + 2 y = 0

and

x - 2 y = 0

.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 2 - Straight Lines>2.8>Q 29

Find the equation of the line through the point of intersection of $x + 2 y = 5$ and $x - 3 y = 7$ , and passing through the point $(0, 0)$ .

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 2 - Straight Lines>2.8>Q 29

Find the equation of the line through the point of intersection of $x + 2 y = 5$ and $x - 3 y = 7$ , and passing through the point $(0, - 1)$ .

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 2 - Straight Lines>2.8>Q 30

Find the equation of the line through the intersection of $5 x - 3 y = 1$ and $2 x + 3 y - 23 = 0$ , and perpendicular to the line whose equation is $x = 0$ .

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 2 - Straight Lines>2.8>Q 30

Find the equation of the line through the intersection of $5 x - 3 y = 1$ and $2 x + 3 y - 23 = 0$ , and perpendicular to the line whose equation is $y = 0$