Practice Quiz on Conic Sections (Parabola, Ellipse, Hyperbola): IITJEE Mains — basics 1

Multiple Choice Questions:

Problem 1:

A line bisecting the ordinate PN of a point P (at^{2}, 2at), t>0, on the parabola y^{2}=4ax is drawn parallel to the axis to meet the curve at Q. If NQ meets the tangent at the vertex at the point T, then the coordinates of T are

(a) (0, \frac{4}{3}at) (b) Slatex (0,2at)$ (c) (\frac{1}{4}at^{2},at) (d) (0,at)

Problem 2:

If P, Q, R are three points on a parabola y^{2}=4ax whose ordinates are in geometrical progression, then the tangents at P and R meet on

(a) the line through Q parallel to x-axis

(b) the line through Q parallel to y-axis

(c) the line through Q to the vertex

(d) the line through Q to the focus.

Problem 3:

The locus of the midpoint 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=-a/2 (c) x=0 (d) x=a/2

Problem 4:

Equation of the locus of the pole with respect to the ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}=1, of any tangent line to the auxiliary circle is the curve \frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}}=\lambda^{2} where

(a) \lambda^{2}=a^{2} (b) \lambda^{2}=\frac{1}{a^{2}} (c) \lambda^{2}=b^{2} (d) \lambda^{2}=\frac{1}{b^{2}}

Problem 5:

The locus of the points of the intersection of the tangents at the extremities of the chords of the ellipse x^{2}+2y^{2}=6, which touch the ellipse x^{2}+4y^{2}=4 is

(a) x^{2}+y^{2}=4 (b) x^{2}+y^{2}=6 (c) x^{2}+y^{2}=9 (d) none of these.

Problem 6:

If an ellipse slides between two perpendicular straight lines, then the locus of its centre is

(a) a parabola (b) an ellipse (c) a hyperbola (d) a circle

Problem 7:

Let P(a\sec{\theta}, b\tan{\theta}) and Q(a\sec{\phi}, b\tan{\phi}) where \theta + \phi=\pi/2, be two points on the hyperbola \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}=1. If (h,k) is the point of intersection of normals at P and Q, then k is equal to

(a) \frac{a^{2}+b^{2}}{a} (b) -(\frac{a^{2}+b^{2}}{a}) (c) \frac{a^{2}+b^{2}}{b} (d) -(\frac{a^{2}+b^{2}}{b})

Problem 8:

A straight line touches the rectangular hyperbola 9x^{2}-9y^{2}=8, and the parabola y^{2}=32x. An equation of the line is

(a) 9x+3y-8=0 (b) 9x-3y+8=0 (c) 9x+3y+8=0 (d) 9x-3y-8=0

There could be multiple answers to this question.

Problem 9:

Two parabolas C and D intersect at the two different points, where C is y=x^{2}-3 and D is y=kx^{2}. The intersection at which the  x-value is positive is designated point A, and x=a at this intersection. The tangent line l at A to the curve D intersects curve C at point B, other than A. If x-value of point B is 1(one), then what is a equal to?

Problem 10:

The triangle formed by the tangent to the parabola y=x^{2} at the point whose abscissa is x_{0}(x_{0} \in [1,2]), the y-axis and the straight line y=a_{0}^{2} has the greatest area if x_{0}= ?. Fill in the question mark!

More later,

Nalin Pithwa.

PS: I think let’s continue this focus on co-ordinate geometry of IITJEE Mains Maths for some more time.


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