The normal to a curve at $$P(x,y)$$ meets the $$x$$-axis at $$G$$. If the distance of $$G$$ from the origin is twice the abscissa of $$P$$, then the curve is a
A
circle
B
hyperbola
C
ellipse
D
parabola
Explanation
Equation of normal at $$P\left( {x,y} \right)$$ is $$Y - y = - {{dx} \over {dy}}\left( {x - x} \right)$$
Coordinate of $$G$$ at $$X$$ axis is $$\left( {X,0} \right)$$ (let)
The equation of a tangent to the parabola $${y^2} = 8x$$ is $$y=x+2$$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is
A
$$(2,4)$$
B
$$(-2,0)$$
C
$$(-1,1)$$
D
$$(0,2)$$
Explanation
Parabola $${y^2} = 8x$$
We know that the locus of point of intersection of two perpendicular tangents to a
parabola is its directrix. Point must be on the directrix of parabola
as equation of directrix $$x + 2 = 0 \Rightarrow x = - 2$$
Hence the point is $$\left( { - 2,0} \right)$$
3
AIEEE 2007
MCQ (Single Correct Answer)
For the Hyperbola $${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$ , which of the following remains constant when $$\alpha $$ varies$$=$$?
A
abscissae of vertices
B
abscissaeof foci
C
eccentricity
D
directrix.
Explanation
Given, equation of hyperbola is $${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$