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1

### IIT-JEE 1995

Subjective
Show that the locus of a point that divides a chord of slope $$2$$ of the parabola $${y^2} = 4x$$ internally in the ratio $$1:2$$ is a parabola. Find the vertex of this parabola.

$$\left( {{2 \over 9},{8 \over 9}} \right)$$
2

### IIT-JEE 1994

Subjective
Through the vertex $$O$$ of parabola $${y^2} = 4x$$, chords $$OP$$ and $$OQ$$ are drawn at right angles to one another . Show that for all positions of $$P$$, $$PQ$$ cuts the axis of the parabola at a fixed point. Also find the locus of the middle point of $$PQ$$.

$${y^2} = 2\left( {x - 4} \right)$$
3

### IIT-JEE 1991

Subjective
Three normals are drawn from the point $$(c, 0)$$ to the curve $${y^2} = x.$$ Show that $$c$$ must be greater than $$1/2$$. One normal is always the $$x$$-axis. Find $$c$$ for which the other two normals are perpendicular to each other.

$$c = {3 \over 4}$$
4

### IIT-JEE 1982

Subjective
$$A$$ is point on the parabola $${y^2} = 4ax$$. The normal at $$A$$ cuts the parabola again at point $$B$$. If $$AB$$ subtends a right angle at the vertex of the parabola. Find the slope of $$AB$$.

$$m = \pm \sqrt 2$$

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Class 12