If the portion of a straight line intercepted | vedclass

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Question

(A) Let the intercepts on the $x$-axis and $y$-axis be $a$ and $b$ respectively. The coordinates of the points are $A(a, 0)$ and $B(0, b)$.Given that

Vedclass · Accepted Answer

$25$

Vedclass · Answer

(A) Let the intercepts on the $x$-axis and $y$-axis be $a$ and $b$ respectively. The coordinates of the points are $A(a, 0)$ and $B(0, b)$.Given that the point $P(2, 3)$ divides the line segment $AB$ in the ratio $2:3$.Using the section formula,the coordinates of $P$ are given by:$P = \left(\frac{2 \cdot 0 + 3 \cdot a}{2 + 3}, \frac{2 \cdot b + 3 \cdot 0}{2 + 3}\right) = \left(\frac{3a}{5}, \frac{2b}{5}\right)$.Equating the coordinates with $(2, 3)$:$\frac{3a}{5} = 2$ $\Rightarrow 3a = 10$ $\Rightarrow a = \frac{10}{3}$.$\frac{2b}{5} = 3$ $\Rightarrow 2b = 15$ $\Rightarrow b = \frac{15}{2}$.The product of the intercepts is $a \cdot b = \left(\frac{10}{3}\right) \cdot \left(\frac{15}{2}\right) = 5 \cdot 5 = 25$.

If the portion of a straight line intercepted between the coordinate axes is divided by the point $(2,3)$ in the ratio $2:3$,then the product of the intercepts made by this line on the axes is

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