Chapter 8

Find the equation of the hyperbola whose focus is \((1,2)\), directrix \(2 x+y=1\) and eccentricity \(\sqrt{3}\).

Show that the equation of the hyperbola having focus \((2,0)\), eccentricity 2 and directrix \(x-y=0\) is \(x^{2}+y^{2}-4 x y+4=0\).

Find the equation of the hyperbola whose centre is \((-3,2)\) and one end of the transverse axis is \((-3,4)\) and eccentricity is \(\frac{5}{2}\).

Find the equation of the hyperbola whose centre is \((3,2)\), one focus is \((5,2)\)and one vertex is \((4,2)\).

Find the equation of the hyperbola whose centre is \((6,2)\), one focus is \((4,2)\)and eccentricity \(2 .\)

Find the centre, eccentricity and foci of hyperbola \(9 x^{2}-16 y^{2}=144\).

Find the centre, foci and eccentricity of \(12 x^{2}-4 y^{2}-24 x+32 y-127=0\).

Find the centre, foci and eccentricity of the hyperbola \(9 x^{2}-4 y^{2}-18 x+16 y-43=0\).

Find the latus rectum of the hyperbola \(4 x-9 y^{2}=36\).

Find the centre, eccentricity and foci of the hyperbola \(x^{2}-2 y^{2}-2 x+8 y-1=0\).

Find the centre, eccentricity, foci and directrix of the hyperbola \(16 x^{2}-9 y^{2}+\) \(32 x+36 y-164=0\)

Tangents are drawn to the hyperbola \(3 x^{2}-2 y^{2}=6\) from the point \(P\) and make \(\theta_{1}, \theta_{2}\) with \(x\) -axis. If the \(\tan \theta_{1} \tan \theta_{2}\) is a constant, prove that locus of \(P\) is \(2 x^{2}-y^{2}=7\)

Find the equation of tangents to the hyperbola \(3 x^{2}-4 y^{2}=15\) which are parallel to \(y=2 x+k\). Find the coordinates of the point of contact.

Tangents are drawn to the hyperbola \(x^{2}-y^{2}=c^{2}\) are inclined at an angle of \(45^{\circ}\), show that the locus of their intersection is \(\left(x^{2}+y^{2}\right)^{2}+4 a^{2}\left(x^{2}-y^{2}\right)=4 a^{4}\).

Prove that the polar of any point on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with respect to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) will touch the ellipse at the other end of the ordinate through the point.

If the polar of points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) with respect to hyperbola are at right angles then show that \(b^{4} x_{1} x_{2}+a^{4} y_{1} y_{2}=0\).

Find the locus of poles of normal chords of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\).

Chords of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) subtend a right angle at one of the vertices. Show that the locus of poles of all such chords is the straight line \(x\left(a^{2}+b^{2}\right)=a\left(a^{2}-b^{2}\right)\)

If chords of the hyperbola are at a constant distance \(k\) from the centre, findthe locus of their poles.

Obtain the locus of the point of intersection of tangents to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) which includes an angle \(\beta\).

If a variable chord of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is a tangent to the circle \(x^{2}+y^{2}=c^{2}\) then prove that the locus of its middle point is \(\left(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right)^{2}=c^{2}\left(\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}\right) .\)

Show that the condition for the line \(x \cos \alpha+y \sin \alpha=\beta\) touches the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is \(a^{2} \cos ^{2} \alpha-b^{2} \sin ^{2} \alpha=p^{2}\)

Prove that the midpoints of the chords of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) parallel to the diameter \(y=m x\) be on the diameter \(a^{2} m y=b^{2} x\).

If the polar of the point \(A\) with respect to a hyperbola passes through another point \(B\), then show that the polar \(B\) passes through \(A\).

If the polars of \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) with respect to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) are at right angles, then prove that \(\frac{x_{1} x_{2}}{y_{1}-y_{2}}+\frac{a^{4}}{b^{4}}=0\).

Prove that the polar of any point on \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with respect to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) touches \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 .\)

Obtain the equation of the chord joining the points \(\theta\) and \(\phi\) on the hyperbola in the form \(\frac{x}{a} \cos \left(\frac{\theta-\phi}{2}\right)-\frac{y}{b} \sin \left(\frac{\theta+\phi}{2}\right)=\cos \left(\frac{\theta+\phi}{2}\right)\). If \(\theta-\phi\) is a constant and equal to \(2 \alpha\), show that \(P Q\) touches the hyperbola \(\frac{x^{2} \cos ^{2} \alpha}{a^{2}}-\frac{y^{2}}{b^{2}}=1\).

If a circle with centre \((3 \alpha, 3 \beta)\) and of variable radius cuts the hyperbola \(x^{2}-y^{2}=9 a^{2}\) at the points \(P, Q, R\) and \(S\) then prove that the locus of the centroid of the triangle \(P Q R\) is \((x-2 \alpha)^{2}-(y-2 \beta)^{2}=a^{2}\).

Show that the locus of the points of intersection of tangents at the extremities of normal chords of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is \(\frac{a^{6}}{x^{2}}-\frac{b^{6}}{v^{2}}=\left(a^{2}+b^{2}\right)^{2}\).

Tangents are drawn from any point on hyperbola \(x^{2}-y^{2}=a^{2}+b^{2}\) to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\). Prove that they meet the axes in conjugate points.

If the chord joining the points \(\alpha\) and \(\beta\) on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is a focal chord then prove that \(\tan \frac{\alpha}{2} \tan \frac{\beta}{2}+\frac{k e-1}{k e+1}=0\) where \(k \neq 1\).

Let the tangent and normal at a point \(P\) on the hyperbola meet the transverse axis in \(T\) and \(G\) respectively, prove that \(C T \cdot C G=a^{2}+b^{2}\).

If the tangent at the point \((h, k)\) to the hyperbola cuts the auxiliary circle in points whose ordinates are \(y_{1}\) and \(y_{2}\) then show that \(\frac{1}{y_{1}}+\frac{1}{y_{2}}=\frac{2}{k}\).

If a line is drawn parallel to the conjugate axis of a hyperbola to meet it and the conjugate hyperbola in the points \(P\) and \(Q\) then show that the tangents at \(P\) and \(Q\) meet on the curve \(\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=\frac{4 b^{4} x^{2}}{x^{2} y^{n}}\).

If the tangent at any point \(P\) on the hyperbola \(\frac{x}{a^{2}}-\frac{y}{b^{2}}=1\) whose centre is \(C\), meets the transverse and conjugate axes in \(T_{1}\) and \(T_{2}\), then prove that (i) \(C N \cdot C T_{1}=a^{2}\) and (ii) \(C M \cdot C T_{2}=-b^{2}\) where \(P M\) and \(P N\) are perpendiculars in the transverse and conjugate axes, respectively.

If \(P\) is the length of the perpendicular from \(C\), the centre of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) on the tangent at a point \(P\) on it and \(C P=r\), prove that \(\frac{a^{2} b^{2}}{p^{2}}=r^{2}+b^{2}-a^{2}\)