Straight Line and Pair of Straight Lines 4 Question 3
3. Show that all chords of curve $3 x^{2}-y^{2}-2 x+4 y=0$, which subtend a right angle at the origin pass through a fixed point. Find the coordinates of the point.
$(1991,4$ M)
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Answer:
Correct Answer: 3. $(1,-2)$
Solution:
- The given curve is
$$ 3 x^{2}-y^{2}-2 x+4 y=0 $$
Let $y=m x+c$ be the chord of curve (i) which subtend right angle at origin. Then, the combined equation of lines joining points of intersection of curve (i) and chord $y=m x+c$ to the origin, can be obtained by the equation of the curve homogeneous, i.e.
$$ \begin{aligned} & 3 x^{2}-y^{2}-2 x \frac{y-m x}{c}+4 y \frac{y-m x}{c}=0 \\ \Rightarrow & 3 c x^{2}-c y^{2}-2 x y+2 m x^{2}+4 y^{2}-4 m x y=0 \\ \Rightarrow & \quad(3 c+2 m) x^{2}-2(1+2 m) y+(4-c) y^{2}=0 \end{aligned} $$
Since, the lines represented are perpendicular to each other.
$\therefore$ Coefficient of $x^{2}+$ Coefficient of $y^{2}=0$
$$ \Rightarrow \quad 3 c+2 m+4-c=0 $$
$$ \Rightarrow \quad c+m+2=0 $$
On comparing with $y=m x+c$
$\Rightarrow \quad y=m x+c$ passes through $(1,-2)$.
