Question

All chords of the curve $3x^2-y^2-2x+4y=0$ which subtend a right angle at the origin, pass through the fixed point

• A. $(1,2)$
• B. $(1,–2)$
• C. $(-1,2)$
• D. $(-1,-2)$

Answer 1

The correct option is B

$(1,–2)$

Explanation for the correct answer:

Step 1: Finding the equation from the chord.

The curve is $3x^2-y^2-2x+4y=0$ --------- (1)

Let the equation, $y=mx+c$ be the chord of the given curve that subtends the right angle at the origin.

$$\begin{gathered} \Rightarrow y=mx+c \\ \Rightarrow y-mx=c\left(1\right) \\ \Rightarrow \frac{y-mx}{c}=1 \end{gathered}$$

Step 2: Substitute the chord equation in 1.

The equation of the intersection of the curve and the chord is

$3x^2-y^2-2x\left(1\right)+4y\left(1\right)=0$ becomes

$$\begin{gathered} \Rightarrow 3x^2-y^2-2x\left(\frac{y-mx}{c}\right)+4y\left(\frac{y-mx}{c}\right)=0 \\ \Rightarrow 3c{x}^2-c{y}^2-2xy+2m{x}^2+4y^2-4mxy=0 \\ \Rightarrow (3c+2m)x^2+(4-c)y^2-2(1+2m)xy=0 \end{gathered}$$

The coefficients of $x^2=a$ and $y^2=b$

Step 3: Finding the coordinate:

The curve and the chord are perpendicular to each other.

So the slope, $\frac{m_1}{m_2}=-1$

Here, $m_1=a,m_2=b$ then

$$\begin{gathered} \Rightarrow \frac{3c+2m}{4-c}=-1 \\ \Rightarrow 3c+2m=-4+c \\ \Rightarrow 3c-c+2m=-4 \\ \Rightarrow 2c+2m=-4 \\ \Rightarrow c+m=-2 \end{gathered}$$

On comparing the equations $-2=m+c$ and $y=mx+c$ with the equation of chord, we get

$x=1andy=-2$

Thus the point $(1,-2)$ is the fixed point through which the curve passes through the origin subtends a right angle.

Hence, option (B) is the correct answer.