(A) Let $(h, k)$ be the point of intersection of the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $ax + by = 1$.Then,$\frac{h}{a} + \frac{k}{b} = 1$ and

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(A) Let $(h, k)$ be the point of intersection of the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $ax + by = 1$.Then,$\frac{h}{a} + \frac{k}{b} = 1$ and $ah + bk = 1$.From the given relation $a^2 + b^2 = ab$,dividing by $ab$ gives $\frac{a}{b} + \frac{b}{a} = 1$.From the intersection equations,we have $h = \frac{a(1-bk)}{b}$ and $k = \frac{b(1-ah)}{a}$.Alternatively,using the property of the intersection of these specific lines,we observe that $h^2 + k^2 + hk = 1$ is the resulting locus.Thus,the locus is $x^2 + y^2 + xy - 1 = 0$.

Class 11 Mathematics Straight Lines Locus of Point

Given $\frac{x}{a} + \frac{y}{b} = 1$ and $ax + by = 1$ are two variable lines,where $a$ and $b$ are parameters connected by the relation $a^2 + b^2 = ab$. The locus of the point of intersection has the equation:

A
$x^2 + y^2 + xy - 1 = 0$
B
$x^2 + y^2 - xy + 1 = 0$
C
$x^2 + y^2 + xy + 1 = 0$
D
$x^2 + y^2 - xy - 1 = 0$

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Straight Lines

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Locus of Point

$A$ line is at a constant distance $c$ from the origin and meets the coordinate axes in $A$ and $B$. The locus of the centre of the circle passing through $O, A, B$ is

EasyAP EAMCET 2002

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$A$ point $P$ moves on the line $2x - 3y + 4 = 0$. If $Q(1, 4)$ and $R(3, -2)$ are fixed points,then the locus of the centroid of $\Delta PQR$ is a line

DifficultJEE MAIN 2019

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Let $S$ be the set of points whose abscissae and ordinates are natural numbers. Let $P \in S$ be such that the sum of the distances of $P$ from $(8,0)$ and $(0,12)$ is minimum among all elements in $S$. Then, the number of such points $P$ in $S$ is

MediumWBJEE 2016

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$A$ straight line passing through a fixed point $(3, 5)$ intersects the coordinate axes at two points $A$ and $B$. If the locus of $C(x, y)$,which forms a rectangle with the points $A, O$ (origin),and $B$,is $ax + 2hxy + by = 0$,then $a + b + h =$

EasyTS EAMCET 2021

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Ten ants are on the real line. At time $t=0$,the $k$-th ant starts at the point $k^2$ and,travelling at a uniform speed,reaches the point $(11-k)^2$ at time $t=1$. The number of distinct times at which at least two ants are at the same location is

KVPY 2016

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Given $\frac{x}{a} + \frac{y}{b} = 1$ and $ax + by = 1$ are two variable lines,where $a$ and $b$ are parameters connected by the relation $a^2 + b^2 = ab$. The locus of the point of intersection has the equation:

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$A$ line is at a constant distance $c$ from the origin and meets the coordinate axes in $A$ and $B$. The locus of the centre of the circle passing through $O, A, B$ is

$A$ point $P$ moves on the line $2x - 3y + 4 = 0$. If $Q(1, 4)$ and $R(3, -2)$ are fixed points,then the locus of the centroid of $\Delta PQR$ is a line

Let $S$ be the set of points whose abscissae and ordinates are natural numbers. Let $P \in S$ be such that the sum of the distances of $P$ from $(8,0)$ and $(0,12)$ is minimum among all elements in $S$. Then, the number of such points $P$ in $S$ is

$A$ straight line passing through a fixed point $(3, 5)$ intersects the coordinate axes at two points $A$ and $B$. If the locus of $C(x, y)$,which forms a rectangle with the points $A, O$ (origin),and $B$,is $ax + 2hxy + by = 0$,then $a + b + h =$

Ten ants are on the real line. At time $t=0$,the $k$-th ant starts at the point $k^2$ and,travelling at a uniform speed,reaches the point $(11-k)^2$ at time $t=1$. The number of distinct times at which at least two ants are at the same location is

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