If the three lines x - 3y = p,ax + 2y = q,and | vedclass

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

Question

(A) The slopes of the three lines are $m_1 = \frac{1}{3}$,$m_2 = -\frac{a}{2}$,and $m_3 = -a$.For the lines to form a right-angled triangle,the produc

Vedclass · Accepted Answer

$a^2 - 9a + 18 = 0$

Vedclass · Answer

(A) The slopes of the three lines are $m_1 = \frac{1}{3}$,$m_2 = -\frac{a}{2}$,and $m_3 = -a$.For the lines to form a right-angled triangle,the product of the slopes of any two perpendicular lines must be $-1$.Case $1$: $m_1 	imes m_2 = -1$ $\Rightarrow \frac{1}{3} 	imes (-\frac{a}{2}) = -1$ $\Rightarrow a = 6$.Case $2$: $m_1 	imes m_3 = -1$ $\Rightarrow \frac{1}{3} 	imes (-a) = -1$ $\Rightarrow a = 3$.Case $3$: $m_2 	imes m_3 = -1$ $\Rightarrow (-\frac{a}{2}) 	imes (-a) = -1$ $\Rightarrow \frac{a^2}{2} = -1$ (No real solution).Thus,$a = 6$ or $a = 3$ are the possible values.Testing the options for $a = 6$ and $a = 3$:For $a = 6$: $6^2 - 9(6) + 18 = 36 - 54 + 18 = 0$.For $a = 3$: $3^2 - 9(3) + 18 = 9 - 27 + 18 = 0$.Both values satisfy the equation $a^2 - 9a + 18 = 0$.

If the three lines $x - 3y = p$,$ax + 2y = q$,and $ax + y = r$ form a right-angled triangle,then:

Similar Questions

The area of the triangle enclosed by the straight lines $x = 0$,$y = 0$ and $x + 2y + 3 = 0$ in sq. units is:

The area of the parallelogram formed by the lines $a_1x + b_1y + c_1 = 0$,$a_1x + b_1y + d_1 = 0$,$a_2x + b_2y + c_2 = 0$,and $a_2x + b_2y + d_2 = 0$ is:

Let the equations of two adjacent sides of a parallelogram $ABCD$ be $2x - 3y = -23$ and $5x + 4y = 23$. If the equation of its one diagonal $AC$ is $3x + 7y = 23$ and the distance of $A$ from the other diagonal is $d$,then $50d^2$ is equal to $........$.

Given two points $A(3, 4)$ and $B(5, -2)$. If $PA = PB$ and the area of $\Delta PAB = 10$ square units,find the coordinates of point $P$.

If a vertex of an equilateral triangle is at the origin and the second vertex is $(4, 0)$,then its third vertex is

Vedclass Test Series

Exam Paper Generator

Online Exam Module