The origin and the points where the line L_1 | vedclass

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

Question

(A) The slope of line $L_2$ is $4$. Since $L_1 \perp L_2$,the slope of $L_1$ is $m = -\frac{1}{4}$.Let the equation of line $L_1$ be $y = -\frac{1}{4}

Vedclass · Accepted Answer

$10 + \sqrt{68}$

Vedclass · Answer

(A) The slope of line $L_2$ is $4$. Since $L_1 \perp L_2$,the slope of $L_1$ is $m = -\frac{1}{4}$.Let the equation of line $L_1$ be $y = -\frac{1}{4}x + k$,which can be written as $x + 4y = 4k$.Let $c = 4k$. The intercepts are $(c, 0)$ and $(0, \frac{c}{4})$.The area of the right-angled triangle $T$ formed by the origin $(0, 0)$ and these intercepts is $\frac{1}{2} 	imes |c| 	imes |\frac{c}{4}| = 8$.$|c^2| = 64$,so $c = \pm 8$.Taking $c = 8$,the vertices are $(0, 0), (8, 0),$ and $(0, 2)$.The lengths of the sides are $8, 2,$ and $\sqrt{8^2 + 2^2} = \sqrt{64 + 4} = \sqrt{68}$.The perimeter is $8 + 2 + \sqrt{68} = 10 + \sqrt{68}$.

The origin and the points where the line $L_1$ intersects the $x$-axis and $y$-axis are vertices of a right-angled triangle $T$ whose area is $8$. Also,the line $L_1$ is perpendicular to the line $L_2: 4x - y = 3$. Then,the perimeter of triangle $T$ is:

Similar Questions

For a point $P(x, y)$ in the plane,let $d_1(P)$ and $d_2(P)$ be the distances of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2 \leq d_1(P)+d_2(P) \leq 4$ is:

If a straight line $L$ perpendicular to the line $3x - 4y = 6$ forms a triangle of area $6$ square units with the coordinate axes,then the minimum perpendicular distance from the point $(1, 1)$ to the line $L$ is

The equation of the line passing through the intersection of $3x - 4y + 1 = 0$ and $5x + y - 1 = 0$ which cuts off equal intercepts on the axes is given by

One vertex of an equilateral triangle is $(2, 3)$ and the line of its opposite side is $x + y = 2$. Find the equations of the other two sides.

The number of integer values of $m$,for which the $x$-coordinate of the point of intersection of the lines $3x + 4y = 9$ and $y = mx + 1$ is also an integer,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module