Straight Line and Pair of Straight Lines 4 Question 1
1. Let $a$ and $b$ be non-zero and real numbers. Then, the equation $\left(a x^{2}+b y^{2}+c\right)\left(x^{2}-5 x y+6 y^{2}\right)=0$ represents
(2008, 3M)
(a) four straight lines, when $c=0$ and $a, b$ are of the same sign
(b) two straight lines and a circle, when $a=b$ and $c$ is of sign opposite to that of $a$
(c) two straight lines and a hyperbola, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
(d) a circle and an ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
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Answer:
Correct Answer: 1. (b)
Solution:
- Let $a$ and $b$ be non-zero real numbers.
Therefore the given equation
$$ \begin{aligned} & \left(a x^{2}+b y^{2}+c\right)\left(x^{2}-5 x y+6 y^{2}\right)=0 \text { implies either } \\ & x^{2}-5 x y+6 y^{2}=0 \\ & \Rightarrow \quad(x-2 y)(x-3 y)=0 \\ & \Rightarrow \quad x=2 y \\ & \text { and } \quad x=3 y \end{aligned} $$
represent two straight lines passing through origin or $a x^{2}+b y^{2}+c=0$ when $c=0$ and $a$ and $b$ are of same signs, then
$$ \begin{aligned} a x^{2}+b y^{2}+c & =0 \\ x & =0 \end{aligned} $$
$$ \text { and } \quad y=0 \text {. } $$
which is a point specified as the origin.
When, $a=b$ and $c$ is of sign opposite to that of $a$, $a x^{2}+b y^{2}+c=0$ represents a circle.
Hence, the given equation,
$$ \left(a x^{2}+b y^{2}+c\right)\left(x^{2}-5 x y+6 y^{2}\right)=0 $$
may represent two straight lines and a circle.
