the normal to the curve x 2 2xy 3y 2 0 at 11

The normal to the curve, $${x^2} + 2xy - 3{y^2} = 0,$$ at $$\left( {1,\,1} \right)$$

A. meets the curve again in the third quadrant.

B. meets the curve again in the fourth quadrant.

C. does not meet the curve again.

D. meets the curve again in the second quadrant.

Answer : meets the curve again in the fourth quadrant.

Solution :
Given curve is
$${x^2} + 2xy - 3{y^2} = 0.....(1)$$
Differentiate w.r.t. $$x,\,\,\,2x + 2x\frac{{dy}}{{dx}} + 2y - 6y\frac{{dy}}{{dx}} = 0$$
$${\left( {\frac{{dy}}{{dx}}} \right)_{\left( {1,\,1} \right)}} = 1$$
Equation of normal at $$\left( {1,\,1} \right)$$ is
$$y = 2 - x.....(2)$$
Solving equation (1) and (2), we get $$x=1 ,\,3$$
Point of intersection $$\left( {1,\,1} \right),\,\left( {3,\, - 1} \right)$$
Normal cuts the curve again in 4th quadrant.

Releted MCQ Question on
Geometry >> Locus

Releted Question 1

The equation $$\frac{{{x^2}}}{{1 - r}} - \frac{{{y^2}}}{{1 + r}} = 1,\,\,\,r > 1$$ represents :

A. an ellipse

B. a hyperbola

C. a circle

D. none of these

View Answer

Releted Question 2

The equation $$2{x^2} + 3{y^2} - 8x - 18y + 35 = k$$ represents :

A. no locus if $$k>0$$

B. an ellipse if $$k<0$$

C. a point if $$k=0$$

D. a hyperbola if $$k>0$$

View Answer

Releted Question 3

If $$a>2b>0$$ then the positive value of $$m$$ for which $$y = mx - b\sqrt {1 + {m^2}} $$ is a common tangent to $${x^2} + {y^2} = {b^2}$$ and $${\left( {x - a} \right)^2} + {y^2} = {b^2}$$ is :

A. $$\frac{{2b}}{{\sqrt {{a^2} - 4{b^2}} }}$$

B. $$\frac{{\sqrt {{a^2} - 4{b^2}} }}{{2b}}$$

C. $$\frac{{2b}}{{a - 2b}}$$

D. $$\frac{b}{{a - 2b}}$$

View Answer

Releted Question 4

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $${y^2} = 4ax$$ is another parabola with directrix :

A. $$x = - a$$

B. $$x = - \frac{a}{2}$$

C. $$x = 0$$

D. $$x = \frac{a}{2}$$

View Answer

The normal to the curve, $${x^2} + 2xy - 3{y^2} = 0,$$ at $$\left( {1,\,1} \right)$$

Releted MCQ Question on
Geometry >> Locus

The equation $$\frac{{{x^2}}}{{1 - r}} - \frac{{{y^2}}}{{1 + r}} = 1,\,\,\,r > 1$$ represents :

The equation $$2{x^2} + 3{y^2} - 8x - 18y + 35 = k$$ represents :

If $$a>2b>0$$ then the positive value of $$m$$ for which $$y = mx - b\sqrt {1 + {m^2}} $$ is a common tangent to $${x^2} + {y^2} = {b^2}$$ and $${\left( {x - a} \right)^2} + {y^2} = {b^2}$$ is :

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $${y^2} = 4ax$$ is another parabola with directrix :

Practice More Releted MCQ Question on
Locus

Practice More MCQ Question on Maths Section

The normal to the curve, $${x^2} + 2xy - 3{y^2} = 0,$$ at $$\left( {1,\,1} \right)$$

Releted MCQ Question on Geometry >> Locus

The equation $$\frac{{{x^2}}}{{1 - r}} - \frac{{{y^2}}}{{1 + r}} = 1,\,\,\,r > 1$$ represents :

The equation $$2{x^2} + 3{y^2} - 8x - 18y + 35 = k$$ represents :

If $$a>2b>0$$ then the positive value of $$m$$ for which $$y = mx - b\sqrt {1 + {m^2}} $$ is a common tangent to $${x^2} + {y^2} = {b^2}$$ and $${\left( {x - a} \right)^2} + {y^2} = {b^2}$$ is :

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $${y^2} = 4ax$$ is another parabola with directrix :

Practice More Releted MCQ Question on Locus

Practice More MCQ Question on Maths Section

Releted MCQ Question on
Geometry >> Locus

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Locus