Name: Special Lines of Triangles
Uploaded: 2019-07-19
Duration: 10 min 17 s
Description: In this video, we shall learn the concept of the special lines in a triangle, such as bisectors, medians, altitudes, and the circumcentre, incentre etc.

Question

A straight line moves in such a way that the length of the perpendicular upon it from the origin is always

p

. Find the locus of the centroid of the triangle which is formed by the line and the axes.

Accepted Answer

Let the equation of the straight line be $y = m x + c$ .

Now, Length of perpendicular upon it from the origin is always $p$

$\Rightarrow p = |\frac{- c}{\sqrt{1 + m^{2}}}|$

$\Rightarrow p = \frac{c}{\sqrt{1 + m^{2}}} . . . . (1)$ (Distance is always positive)

Now, the $x & y$ intercept of the line are $(\frac{- c}{m}, 0) & (0, c)$ respectively.

Now, the coordinates of the triangle formed by $(0, 0), (\frac{- c}{m}, 0)$ and $(0, c)$ are $(\frac{0 + (\frac{- c}{m}) + 0}{3}, \frac{0 + 0 + c}{3})$

$\Rightarrow (\frac{- c}{3 m}, \frac{c}{3})$

Let $x_{1} = \frac{- c}{3 m}, y_{1} = \frac{c}{3} . . . . (2)$

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

$x_{1} = \frac{- 1}{3 m} (p \sqrt{1 + m^{2}}), y_{1} = \frac{p}{3} \sqrt{1 + m^{2}}$

$\Rightarrow \frac{- 3 m x_{1}}{p} = \sqrt{1 + m^{2}}, \frac{3 y_{1}}{p} = \sqrt{1 + m^{2}}$

$\Rightarrow \frac{9 m^{2} {x_{1}}^{2}}{p^{2}} = 1 + m^{2}, \frac{9 {y_{1}}^{2}}{p^{2}} = 1 + m^{2}$

Simplifying the above equations we get,

$\frac{9 {y_{1}}^{2}}{p^{2}} = 1 + \frac{p^{2}}{9 {x_{1}}^{2} - p^{2}}$

$\Rightarrow \frac{9 {y_{1}}^{2}}{p^{2}} = \frac{9 {x_{1}}^{2}}{9 {x_{1}}^{2} - p^{2}}$

$\Rightarrow \frac{1}{{y_{1}}^{2}} + \frac{1}{{x_{1}}^{2}} = \frac{9}{p^{2}}$

So, the locus of the centroid of the triangle which is formed by the line and the axes is $\frac{1}{x^{2}} + \frac{1}{y^{2}} = \frac{9}{p^{2}}$ .

A straight line moves in such a way that the length of the perpendicular upon it from the origin is always $p$ . Find the locus of the centroid of the triangle which is formed by the line and the axes.

Important Questions on Coordinates and Straight Lines

Learn and Prepare for any exam you want !

A straight line moves in such a way that the length of the perpendicular upon it from the origin is always p. Find the locus of the centroid of the triangle which is formed by the line and the axes.

Important Questions on Coordinates and Straight Lines

Learn and Prepare for any exam you want !

A straight line moves in such a way that the length of the perpendicular upon it from the origin is always $p$ . Find the locus of the centroid of the triangle which is formed by the line and the axes.