A variable straight-line L with negative slop | vedclass

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

Question

(A) Let the coordinates of $A$ be $(a, 0)$ and $B$ be $(0, b)$, where $a, b > 0$.The equation of the line passing through $(a, 0)$ and $(0, b)$ is $\f

Vedclass · Accepted Answer

$25$

Vedclass · Answer

(A) Let the coordinates of $A$ be $(a, 0)$ and $B$ be $(0, b)$, where $a, b > 0$.The equation of the line passing through $(a, 0)$ and $(0, b)$ is $\frac{x}{a} + \frac{y}{b} = 1$.Since the line passes through $(4, 9)$, we have $\frac{4}{a} + \frac{9}{b} = 1$.We want to minimize $S = a + b$.From the equation $\frac{4}{a} + \frac{9}{b} = 1$, we can express $b$ in terms of $a$: $\frac{9}{b} = 1 - \frac{4}{a} = \frac{a-4}{a}$, so $b = \frac{9a}{a-4}$.Thus, $S(a) = a + \frac{9a}{a-4}$.To find the minimum, we differentiate $S(a)$ with respect to $a$: $S'(a) = 1 + \frac{9(a-4) - 9a}{(a-4)^2} = 1 - \frac{36}{(a-4)^2}$.Setting $S'(a) = 0$, we get $(a-4)^2 = 36$, so $a-4 = 6$ (since $a > 4$), which gives $a = 10$.Then $b = \frac{9(10)}{10-4} = \frac{90}{6} = 15$.The minimum value is $S = a + b = 10 + 15 = 25$.

$A$ variable straight-line $L$ with negative slope passes through the point $(4,9)$ and cuts the positive coordinate axes in $A$ and $B$. If $O$ is the origin, then the minimum value of $OA+OB$ is

Similar Questions

The locus of a point which moves such that the area of the triangle formed by it with the vertices $(1, 2)$ and $(-2, 5)$ is $8$ sq. units is/are

If the sum of the distances of a point from two perpendicular lines in a plane is $1$,find its locus.

If the sum of the distances of a point from two perpendicular lines in a plane is $1$ unit, then its locus is

If $M$ is the foot of the perpendicular drawn from the origin $O$ to a variable line $L$ passing through a fixed point $Q(a, b)$,then the locus of the mid-point of $OM$ is

The point $P$ is equidistant from $A(1, 3)$,$B(-3, 5)$,and $C(5, -1)$. Then $PA$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module