Question:

If the medians ADAD and BEBE of the triangle with vertices A(0,b),B(0,0),C(a,0)A(0, b), B(0, 0), C(a, 0) are mutually perpendicular, then

Updated On: May 11, 2024
  • b=2ab = \sqrt{2a}
  • a=±2ba = \pm \sqrt{2b}
  • b=−2ab = - \sqrt{2a}
  • b=ab = a
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The Correct Option is B

Solution and Explanation

ADAD and BEBE are perpendicular.



DD and EE are the mid points of BCBC and ACAC respectively.
∴ \therefore Coordinates of
D=(0+a2,0+a2)=(a2,0)D = \left( \frac{0+a}{2} , \frac{0+a}{2} \right) = \left( \frac{a}{2} , 0\right)
AD⊥BEAD \bot BE
∴ \therefore Slope of AD ×\times Slope of BE = -1
(0−ba2−0)×(b2−0a2−0)=−1\left(\frac{0-b}{\frac{a}{2} -0}\right)\times\left(\frac{\frac{b}{2} -0}{\frac{a}{2} -0}\right) = -1
⇒b22=a24⇒a2=2b2⇒a=±2b.\Rightarrow \frac{b^{2}}{2} = \frac{a^{2}}{4} \Rightarrow a^{2} = 2b^{2} \Rightarrow a = \pm\sqrt{2} b.
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Concepts Used:

Straight lines

A straight line is a line having the shortest distance between two points. 

A straight line can be represented as an equation in various forms,  as show in the image below:

 

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y0 = m (x – x0)

2. Two – Point Form

Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2)  are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes

The slope of P2P = The slope of P1P2 , i.e.

y−y1x−x1=y2−y1x2−x1\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}

Hence, the equation becomes:

y - y1 =y2−y1x2−x1(x−x1) \frac{y_2-y_1}{x_2-x_1} (x-x1)

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c