Chapter 6

Show that two tangents can be drawn from a given point to a parabola. If the tangents make angles \(\theta_{1}\) and \(\theta_{2}\) with \(x\) axis such that (i) \(\tan \theta_{1}+\tan \theta_{2}\) is a constant show that the locus of point of intersection of tangents is a straight line through the vertex of a parabola. (ii) if \(\tan \theta_{1} \cdot \tan \theta_{2}\) is a constant show that the locus of the point of intersecting is a straight line. (iii) if \(\theta_{1}+\theta_{2}\) is a constant show that the locus of the point of intersection of tangents is a straight line through the focus. (iv) if \(\theta_{1}\) and \(\theta_{2}\) are complementary angles then the locus of point of intersection is the straight line \(x=a\).

Find the locus of point of intersection of tangents to the parabola \(y^{2}=4 a x\) which includes an angle of \(\frac{\pi}{3}\).

Show that the locus of the poles of chords of the parabola \(y^{2}=4 a x\) which subtends an angle of \(45^{\circ}\) at the vertex is the curve \((x+a)^{2}=4\left(y^{2}-4 a x\right)\).

Show that the locus of poles of all tangents to the parabola \(y^{2}=4 a x\) with respect to the parabola \(y^{2}=4 b x\) is the parabola \(a y^{2}=4 b^{2} x\).

Show that if tangents be drawn to the parabola \(y^{2}=4 a x\) from any point on the straight line \(x+4 a=0\), the chord of contact subtends a right angle at the vertex of the parabola.

Perpendiculars are drawn from points on the tangent at the vertex on their polars with respect to the parabola \(y^{2}=4 a x\). Show that the locus of the foot of the perpendicular is a circle centre at \((a, 0)\) and radius \(a\).

Show that the locus of poles with respect to the parabola \(y^{2}=4 a x\) of tangents to the circle \(x^{2}+y^{2}=4 a^{2}\) is \(x^{2}-y^{2}=4 a^{2}\).

A point \(P\) moves such that the line through the perpendicular to its polar with respect to the parabola \(y^{2}=4 a x\) touches the parabola \(x^{2}=4 b y .\) Show that the locus of \(P\) is \(2 a x+b y+4 a^{2} x=0\).

If a chord of the parabola \(y^{2}=4 a x\) subtends a right angle at its focus, show that the locus of the pole of this chord with respect to the given parabola is \(x^{2}-y^{2}+6 a x+a^{2}=0 .\)

Show that the locus of poles of all chords of the parabola \(y^{2}=4 a x\) which are at a constant distance \(d\) from the vertex is \(d^{2} y^{2}+4 a^{2}\left(d^{2}-x^{2}\right)=0\).

Show that the locus of poles of the focal chords of the parabola \(y^{2}=4 a x\) is \(x+a=0\).

Prove that the polar of any point on the circle \(x^{2}+y^{2}-2 a x-3 a^{2}=0\) with respect to the circle \(x^{2}+y^{2}+2 a x-3 a^{2}=0\) touches the parabola \(y^{2}=4 a x\).

Show that the locus of the poles with respect to the parabola \(y^{2}=4 a x\) of the tangents to the curve \(x^{2}-y^{2}=a\) is the ellipse \(4 x^{2}+y^{2}=4 a x\).

\(P\) is a variable point on the line \(y=b\), prove that the polar of \(P\) with respect to the parabola \(y^{2}=4 a x\) is a fixed directrix.

Tangents are drawn to the parabola \(y^{2}=4 a x\) from a point \((h, k)\). Show that the area of the triangle formed by the tangents and the chord of contact is \(\frac{\left(k^{2}-4 a h\right)^{3 / 2}}{a} .\)

Prove that the length of the chord of contact of the tangents drawn from the point \(\left(x_{1}, y_{1}\right)\) to the parabola \(y^{2}=4 a x\) is \(\frac{1}{a} \sqrt{y_{1}^{2}+4 a^{2}}\left(y_{1}^{2}-4 a x_{1}\right)\). Hence show that one of the triangles formed by these tangents and their chord of contact is \(\frac{1}{2 a}\left(y_{1}^{2}-4 a x_{1}\right)^{3 / 2}\)

Prove that the tangent to a parabola and the perpendicular to it from its focus meet on the tangent at the vertex.

Show that a portion of a tangent to a parabola intercepted between directrix and the curve subtends a right angle at the focus.

Show that a portion of a tangent to a parabola intercepted between directrix and the curve subtends a right angle at the focus.

The tangent to the parabola \(y^{2}=4 a x\) make angles \(\theta_{1}\) and \(\theta_{2}\) with the axIs. Show that the locus of the point of intersection such that \(\cot \theta_{1}+\cot \theta_{2}=c\) is \(y=a c .\)

Prove that the equation of the parabola whose vertex and focus on \(x\) -axis at distances \(4 a\) and \(5 a\) from the origin respectively \((a>0)\) is \(y^{2}=4 a(x-4 a)\). Also obtain the equation to the tangent to this curve at the end of latus rectum in the first quadrant.

Chords of a parabola are drawn through a fixed point. Show that the locus of the middle points is another parabola.

Through each point of the straight line \(x-m y=h\) is drawn a chord of the parabola \(y^{2}=4 a x\) which is bisected at the point. Prove that it always touches the parabola \((y+2 a m)^{2}=8 a x h\).

Two tangents drawn from a point to the parabola make angles \(\theta_{1}\) and \(\theta_{2}\) with the \(x\) -axis. Show that the locus of their point of intersection if \(\tan ^{2} \theta_{1}+\) \(\tan ^{2} \theta_{2}=c\) is \(y^{2}-c x^{2}=2 a x\).

Chords of the parabola \(y^{2}=4 a x\) are drawn through a fixed point \((h, k)\). Show that the locus of the midpoint is a parabola whose vertex is \(\left(h-\frac{k^{2}}{8 a}, \frac{k}{2}\right)\) and latus rectum is \(2 a .\)

Show that the locus of the middle points of a system of parallel chords of a parabola is a line which is parallel to the axis of the parabola.

Show that the locus of the midpoints of chords of the parabola which subtends a constant angle \(\alpha\) at the vertex is \(\left(y^{2}-2 a x-8 a^{2}\right)^{2} \tan ^{2} \alpha=16 a^{2}\left(4 a x-y^{2}\right)\).