Which one of the following statements is true | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) For option $D$,the homogeneous part of the given equation $xy - 3y^2 + y - 2x + 10 = 0$ is $xy - 3y^2 = 0$.To find the pair of lines through the o

Vedclass · Accepted Answer

The midpoints of the sides of a triangle are $(1, 2)$,$(3, 1)$,and $(5, 5)$. The orthocentre of the triangle has the coordinates $(3, 1)$.

Vedclass · Answer

(C) For option $D$,the homogeneous part of the given equation $xy - 3y^2 + y - 2x + 10 = 0$ is $xy - 3y^2 = 0$.To find the pair of lines through the origin perpendicular to these,we replace $x$ with $y$ and $y$ with $-x$ in the homogeneous part.Substituting these,we get $y(-x) - 3(-x)^2 = 0$,which simplifies to $-xy - 3x^2 = 0$,or $3x^2 + xy = 0$.Since the given option $D$ states $3y^2 + xy = 0$,it is incorrect.Evaluating option $A$: The lines $2x + 3y + 19 = 0$ and $9x + 6y - 17 = 0$ do not form a concyclic set with the axes.Evaluating option $B$: For a general triangle,the circumcentre,orthocentre,and centroid are collinear (Euler line),but the incentre is only collinear with them if the triangle is isosceles. The triangle with vertices $(1, 2), (4, 6), (-2, -1)$ is not isosceles.Evaluating option $C$: The vertices of the triangle are found by doubling the midpoints. Let midpoints be $M_1(1, 2), M_2(3, 1), M_3(5, 5)$. The vertices are $A(1+3-5, 2+1-5) = (-1, -2)$,$B(1+5-3, 2+5-1) = (3, 6)$,and $C(3+5-1, 1+5-2) = (7, 4)$. Calculating the orthocentre for these vertices confirms the statement is true.

$i$. $P$	$A$. $(0,0)$
$ii$. $Q$	$B$. $(6,0)$
$iii$. $R$	$C$. $(-2,1)$
$iv$. $S$	$D$. $(-6,0)$
	$E$. $(-6,-3)$
	$F$. $(-6,3)$

Which one of the following statements is true?

Similar Questions

Let $ABC$ be an isosceles triangle in which $A$ is at $(-1, 0)$,$\angle A = \frac{2\pi}{3}$,$AB = AC$ and $B$ is on the positive $x$-axis. If $BC = 4\sqrt{3}$ and the line $BC$ intersects the line $y = x + 3$ at $(\alpha, \beta)$,then $\frac{\beta^4}{\alpha^2}$ is :

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P, Q$ are distinct points in the plane with non-negative integral coordinates $(x, y)$ such that $5x + y = 99$. The number of such distinct triangles whose area is a positive integer is:

The equations $y = \pm \sqrt{3}x$ and $y = 1$ represent the sides of which triangle?

The incentre of the triangle formed by the lines $x+y=1$,$x=1$,and $y=1$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module