(A) The line equation is $px + qy = r$. Dividing by $r$, we get $\frac{x}{r/p} + \frac{y}{r/q} = 1$.The intercepts on the axes are $a = \frac{r}{p}$ a

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$p, 3\sqrt{3}r, q$ are in $G.P.$

(A) The line equation is $px + qy = r$. Dividing by $r$, we get $\frac{x}{r/p} + \frac{y}{r/q} = 1$.The intercepts on the axes are $a = \frac{r}{p}$ and $b = \frac{r}{q}$.The area of the triangle formed is $\frac{1}{2} \times |a| \times |b| = \frac{1}{2} \times \frac{r}{p} \times \frac{r}{q} = \frac{r^2}{2pq}$.Given area is $\frac{1}{54}$, so $\frac{r^2}{2pq} = \frac{1}{54}$.This simplifies to $2pq = 54r^2$, or $pq = 27r^2$.This can be written as $pq = (3\sqrt{3}r)^2$.Therefore, $p, 3\sqrt{3}r, q$ are in $G.P.$

Class 11 Mathematics Straight Lines Mix Examples-Straight Line

Difficult

The area of the triangle formed by the coordinate axes and the line $px + qy = r$ (where $p, q$ and $r$ are positive real numbers) is $\frac{r^2}{2pq}$. If this area is $\frac{1}{54}$ sq. units, then which of the following is true?

MHT CET 2026 All MHT CET

A
$p, 3\sqrt{3}r, q$ are in $G.P.$
B
$p^2 + q^2 = r^2$
C
$p, r, q$ are in $G$.$P$.
D
$p, 27r, q$ are in $G$.$P$.

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The area of the triangle formed by the coordinate axes and the line $px + qy = r$ (where $p, q$ and $r$ are positive real numbers) is $\frac{r^2}{2pq}$. If this area is $\frac{1}{54}$ sq. units, then which of the following is true?

Similar Questions

The equation of one of the sides of an isosceles right-angled triangle,whose hypotenuse is $3x + 4y = 4$ and the vertex opposite to the hypotenuse is $(2, 2)$,is:

The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line $L: 9x + 5y = 45$ between the axes. If $m_1$ and $m_2$ are the slopes of the lines $l_1$ and $l_2$,then the point of intersection of the line $y = (m_1 + m_2)x$ with $L$ lies on

Let $A(-1, 1)$,$B(3, 4)$,and $C(2, 0)$ be three given points. $A$ line $y = mx$,$m > 0$,intersects lines $AC$ and $BC$ at points $P$ and $Q$ respectively. Let $A_1$ and $A_2$ be the areas of $\Delta ABC$ and $\Delta PQC$ respectively,such that $A_1 = 3A_2$. Then the value of $m$ is equal to:

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