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1

### IIT-JEE 2007

Subjective
Match the statements in Column $$I$$ with the properties in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS$$.

Column $$I$$
(A) Two intersecting circles
(B) Two mutually external circles
(C) Two circles, one strictly inside the other
(D) Two branches vof a hyperbola

Column $$II$$
(p) have a common tangent
(q) have a common normal
(r) do not have a common tangent
(s) do not have a common normal

$$\left( A \right) - p,q;\,$$ $$\,\,\left( B \right) - p,q;$$ $$\left( C \right) - q,r;\,$$ $$\left( D \right) - q,r$$
2

### IIT-JEE 2005

Subjective
Find the equation of the common tangent in $${1^{st}}$$ quadrant to the circle $${x^2} + {y^2} = 16$$ and the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over 4} = 1$$. Also find the length of the intercept of the tangent between the coordinate axes.

$$y = - {2 \over {\sqrt 3 }}x + 4\sqrt {{7 \over 3}} ,\,\,\,\,\,{{14} \over {\sqrt 3 }}$$
3

### IIT-JEE 2005

Subjective
Tangents are drawn from any point on the hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ to the circle $${x^2} + {y^2} = 9$$.Find the locus of mid-point of the chord of contact.

$${{{x^2}} \over 9} - {{{y^2}} \over 4} = {\left( {{{{x^2} + {y^2}} \over 9}} \right)^2}$$
4

### IIT-JEE 2004

Subjective
Tangent is drawn to parabola $${y^2} - 2y - 4x + 5 = 0$$ at a point $$P$$ which cuts the directrix at the point $$Q$$. $$A$$ point $$R$$ is such that it divides $$QP$$ externally in the ratio $$1/2:1$$. Find the locus of point $$R$$

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

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