Name: Angle Bisectors of Lines
Uploaded: 2019-07-19
Duration: 8 min 21 s
Description: How do you find the equation of an angle bisector of two lines? Well, watch this video to get all the answers related to this! Have fun learning!

Question

Find the coordinates of the point which is at a distance of

2

units from

(5, 4)

and

10

units from

(11, - 2)

.

Accepted Answer

Let $P (x, y)$ be the required point.

Let $A (5, 4)$ and $B (11, - 2)$

According to the question, $P A = 2, P B = 10$

Now, $P A^{2} = 4$

$\Rightarrow (x - 5)^{2} + (y - 4)^{2} = 4$

$\Rightarrow x^{2} - 10 x + 25 + y^{2} - 8 y + 16 = 4$

$\Rightarrow x^{2} + y^{2} - 10 x - 8 y + 41 - 4 = 0$

$\Rightarrow x^{2} + y^{2} - 10 x - 8 y + 37 = 0 \to (1)$

Also, $P B = 10$

$\Rightarrow P B^{2} = 100$

$\Rightarrow (x - 11)^{2} + (y + 2)^{2} = 100$

$\Rightarrow x^{2} - 22 x + 121 + y^{2} + 4 y + 4 = 100$

$\Rightarrow x^{2} + y^{2} - 22 x + 4 y + 25 = 0 \to (2)$

Now subtract $(1)$ from $(2)$ , we get

$\Rightarrow - 12 x + 12 y - 12 = 0$

$\Rightarrow 12 x - 12 y + 12 = 0$

$\Rightarrow x - y + 1 = 0$

$\Rightarrow x = y - 1 \to (3)$

Now substitute $(3)$ in $(1)$ , we get

$\Rightarrow (y - 1)^{2} + y^{2} = 10 (y - 1) + 8 y - 37$

$\Rightarrow 2 y^{2} - 2 y + 1 = 10 y - 10 + 8 y - 37$

$\Rightarrow 2 y^{2} - 20 y + 48 = 0$

$\Rightarrow y^{2} - 10 y + 24 = 0$

$\Rightarrow y^{2} - 6 y - 4 y + 24 = 0$

$\Rightarrow (y - 6) (y - 4) = 0$

$\Rightarrow y = 6$ or $y = 4$

If $y = 6$ , from $(3)$ , $x = 6 - 1 = 5$

If $y = 4$ , from $(3)$ , $x = 4 - 1 = 3$

Hence, the required co-ordinates are $(5, 6)$ or $(3, 4)$ .

Find the coordinates of the point which is at a distance of $2$ units from $(5, 4)$ and $10$ units from $(11, - 2)$ .

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Find the coordinates of the point which is at a distance of 2 units from (5,4) and 10 units from 11,-2.

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Find the coordinates of the point which is at a distance of $2$ units from $(5, 4)$ and $10$ units from $(11, - 2)$ .