Name: Sum and Product of the Slopes of Pair of Straight Lines
Uploaded: 2019-07-19
Duration: 5 min 24 s
Description: A combined equation of a pair of straight lines is obtained by multiplying the equations of two straight lines. Let's learn about their sum and product of sl...

Question

Find the length of the perpendicular and the co-ordinates of the foot of the perpendicular from

(- 10, - 2)

to the line

x + y - 2 = 0 .

Accepted Answer

Given line is $x + y - 2 = 0$

point $(- 10, - 2)$

Let $C D = x + y - 2$

$A = (- 10, - 2)$

Length of perpendicular $A B = |\frac{a x_{1} + b y_{1} + c}{\sqrt{a^{2} + b^{2}}}|$

Here $a = 1, b = 1, c = - 2$

$(x_{1}, y_{1}) = (- 10, - 2)$

$A B = |\frac{1 (- 10) + 1 (- 2) + (- 2)}{\sqrt{1^{2} + 1^{2}}}|$

$= |\frac{- 10 - 2 - 2}{\sqrt{2}}|$

$= |\frac{- 14}{\sqrt{2}}|$

$= \frac{14}{\sqrt{2}}$

$= \frac{7 \times 2}{\sqrt{2}} = \frac{7 \times \sqrt{2} \times \sqrt{2}}{\sqrt{2}}$

$= 7 \sqrt{2}$

Length of the perpendicular $= 7 \sqrt{2}$

Now we have to find the coordinates of foot of the perpendicular from $(- 10, - 2)$

Let $x + y - 2 = 0 . . . . . (1)$

Equation of $A B$ is perpendicular to $C D$

So, equation of $A B$ is $- x + y + k = 0 . . . . (2)$

Substitute $(- 10, - 2)$ in equation $(2)$

$- (- 10) - 2 + k = 0$

$10 - 2 + k = 0$

$k = - 8$

Substitute $k = - 8$ in equation $(2)$

$A B \Rightarrow - x + y + k = 0$

$- x + y - 8 = 0$

$x - y + 8 = 0 . . . . . (3)$

Equation of $A B \Rightarrow x - y + 8 = 0$

Adding equation $(1) & (3)$

$\Rightarrow x - y + 8 + x + y - 2 = 0$

$\Rightarrow 2 x + 6 = 0$

$\Rightarrow 2 x = - 6$

$\Rightarrow x = - 3$

Put $x = - 3$ in equation $x - y + 8 = 0$

$\Rightarrow - 3 - y + 8 = 0$

$\Rightarrow - y + 5 = 0$

$\Rightarrow - y = - 5$

$\Rightarrow y = 5$

Coordinates of the foot of the perpendicular from $(- 10, - 2)$ to the line is $x = - 3, y = 5$

$∴ (- 3, 5)$

Find the length of the perpendicular and the co-ordinates of the foot of the perpendicular from $(- 10, - 2)$ to the line $x + y - 2 = 0 .$

Important Questions on Two Dimensional Analytical Geometry

Learn and Prepare for any exam you want !

Find the length of the perpendicular and the co-ordinates of the foot of the perpendicular from (−10,−2) to the line x+y−2=0.

Important Questions on Two Dimensional Analytical Geometry

Learn and Prepare for any exam you want !

Find the length of the perpendicular and the co-ordinates of the foot of the perpendicular from $(- 10, - 2)$ to the line $x + y - 2 = 0 .$