## 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.