Conics: Homework for IITJEE Mains: Hausaufgabe Grundlagen konischen!

Aber das ist English ! 🙂

Question 1:

Find the equation to that tangent to the parabola y^{2}=7x which is parallel to the straight line 4y-x+3=0. Find also its point of contact.

Question 2:

If P, Q and R are three points on the parabola y^{2}=4ax whose coordinates are in geometric progression, prove that the tangents at P and R meet on the ordinate of Q.

Question 3:

PNP^{'} is a double ordinate of the parabola y^{2}=4ax. Prove that the locus of the point of intersection of the normal at P and straight line through P^{'} parallel to the axis is the parabola y^{2}=4a(x-4a).

Question 4:

Show that in a parabola, the length of the focal chord varies inversely as the square of the distance of the vertex of the parabola from the focal chord.

Question 5:

Prove that the equation y^{2}+2ax+2by+c=0 represents a parabola whose axis is parallel to the x-axis.. Find its vertex.

Question 6:

If the line y=3x+1 touches the parabola y^{2}=4ax, find the length of the latus rectum.

Question 7a:

Prove that the circle described on any focal chord of a parabola as the diameter touches the directrix of the parabola.

Question 7b:

Show that the locus of the point, such that two of the normals drawn from it to the parabola y^{2}=4ax coincide is 27ay^{2}=4(x-2a)^{3}.

Question 7c:

If the normals at three points A, B and C on the parabola y^{2}=4ax pass through the point S(h,k) and cut the axis of the parabola in P, Q and R so that OP, OQ, OR are in AP, O being the vertex of the parabola, prove that the locus of the point S is 27ay^{}2=2(x-2a)^{2}.

Question 8:

If P, Q and R be three conormal points on the parabola y^{2}=4ax, the normals at which pass through the point T, and S is the focus of the parabola, then prove that SP.SQ.SR=a.ST^{2}

Question 9:

The tangents at P and Q to the parabola y^{2}=4ax meet in T and the corresponding normals meet in R. If the locus of T is a straight line parallel to the axis of the parabola, prove that the locus of R is a straight line normal to the parabola.

Question 10a:

A variable chord PQ of the parabola y^{2}=4x is drawn parallel to the line y=x. If the parameters of the points P and Q on the parabola be t_{1} and t_{2}, then t_{1}+t_{2}=2. Also, show that the locus of the point of intersection of the normals at P and Q is 2x-y=12, which is itself a normal to the parabola.

Question 10b:

PQ is a chord of the parabola y^{2}=36x whose right bisector meets the axis in M and the ordinate of the mid-point of PQ meets the axis in L. Show that LM is constant and find LM.

Question 11:

Prove that the locus of a point P such that the slopes m_{1}, m_{2}, m_{3} of the three normals drawn to the parabola y^{2}=4x from P be connected by the relation \arctan{m_{1}^{2}}+\arctan{m_{2}^{2}} + \arctan{m_{3}^{2}=\alpha} is x^{2}\tan{\alpha}-y^{2}+2(1-2\tan{\alpha})x+(3\tan{\alpha}-4)=0.

Question 12a:

If the perpendicular drawn from P on the polar of P with respect to the parabola, y^{2}=4by, prove that the locus of P is the straight line 2ax+by+4a^{2}=0.

Question 12b:

Prove that the locus of the poles of the tangent to the parabola y^{2}=4ax w.r.t. the circle x^{2}+y^{2}=2ax is x^{2}+y^{2}=ax.

Question 13:

Tangents are drawn to the parabola y^{2}=4ax at the points P and Q whose inclination to the axis are \theta_{1}, \theta_{2}. If A be the vertex of the parabola and the circles on AP and AQ as diameters intersect in R and AR be inclined at an angle \phi to the axis, then prove that \cot{\theta_{1}}+\cot_{\theta_{2}}+2\tan_{\phi}=0.

Question 14:

Through the vertex O of a parabola y^{2}=4x chords OP and OQ are drawn at right angles to one another. Prove 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.

Question 15a:

Prove that the area of a triangle inscribed in a parabola is twice the area of the triangle formed by the tangents at the vertices of the triangle.

Question 15b:

Normals are drawn to the parabola y^{2}=4ax at points A, B, and C whose parameter are t_{1}, t_{2}, t_{3} respectively. If these normals enclose a triangle PQR, then prove that its area is \frac{a^{2}}{2}(t_{1}-t_{2})(t_{2}-t_{3})(t_{3}-t_{1})(t_{1}+t_{2}+t_{3})^{2}. Also, prove that \triangle PQR=\triangle ABC (t_{1}+t_{2}+t_{3})^{2}.

Question 16:

Find the locus of the middle points of the chords of the ellipse (\frac{x^{2}}{a^{2}})+(\frac{y^{2}}{b^{2}})=1 which are drawn through the positive end of the minor axis.

Question 17:

Prove that the sum of the squares of the perpendiculars on any tangent to the ellipse (\frac{x^{2}}{a^{2}})+(\frac{y^{2}}{b^{2}})=1 from the points on the minor axis, each at a distance \sqrt{a^{2}-b^{2}} from the centre, is 2a^{2}.

Question 18:

If P(a\cos{\alpha}, b\sin{\alpha}) and Q(a\cos{\beta}, b\sin{\beta}) are two variable points on the ellipse (\frac{x^{2}}{a^{2}})+(\frac{y^{2}}{b^{2}})=1 such that \alpha+\beta=2\gamma (some constant), then prove that the tangent at (a\cos{\gamma},b\sin{\gamma}) is parallel to PQ.

Question 19:

P is a variable point on the ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}=1 whose foci are the points S_{1}, S_{2}, and the eccentricity is e. Prove that the locus of incentre of \triangle PS_{1}S_{2} is an ellipse whose eccentricity is \sqrt{\frac{2e}{1+e}}.

Question 20:

Consider the family of circles x^{2}+y^{2}=r^{2}, 2<r<5. If in the first quadrant, the common tangent to a circle of this family and the ellipse 4x^{2}+25y^{2}=100 meets the coordinate axes at A and B, then find the equation of the locus of the mid-point of AB.

Question 21:

Let P be a point on the ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}=1, 0<b<a. Let the line parallel to y-axis passing through P meet the circle x^{2}+y^{2}=a^{2} at the point Q such that P and Q are on the same side of the x-axis. For two positive real numbers, r and s, find the locus of the point R on PQ such that PR:RQ=r:s as P varies over the ellipse.

Question 22:

If the eccentric angles of points P and Q on the ellipse be \theta and \frac{\pi}{2} + \theta and \alpha be the angle between the normals at P and Q, then prove that the eccentricity e is given by 2\sqrt{1-e^{2}}=e^{2}=e^{2}\sin^{2}{2\theta}\tan{\alpha}.

Question 23:

A series of hyperbolas are such that the length of their transverse axis is 2a. Prove that the locus of a point P on each, such that its distance from transverse axis is equal to its distance from an asymptote is the curve:

(x^{2}-y^{2})^{2}=4x^{2}(x^{2}-a^{2}).

Question 24:

A variable line of slope 4 intersects the hyperbola xy=1 at two points. Find the locus of the point which divides the line segment between these two points in the ratio 1:2.

Question 25:

If the tangent at the point (p,q) on the hyperbola \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}=1 cuts the auxillary circle in points whose coordinates are y_{1} and y_{2}, then show that q is harmonic mean of y_{1} and y_{2}.

Question 26:

Show that the locus of poles with respect to the parabola y^{2}=4ax of the tangents to the hyperbola x^{2}-y^{2}=a^{2} to the ellipse 4x^{2}+y^{2}=4a^{2}.

Question 27:

The point P on the hyperbola with focus S is such that the tangent at P, the latus rectum through S and one asymptote are concurrent. Prove that SP is parallel to other asymptote.

Question 28:

If a triangle is inscribed in a rectangular hyperbola, prove that the orthocentre of the triangle lies on the curve.

Question 29:

A series of chords of the hyperbola \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}=1 touch the circle on the line joining the foci as diameter. Show that the locus of the poles of these chords with respect to the hyperbola is \frac{x^{2}}{a^{4}} - \frac{y^{2}}{b^{4}} = \frac{1}{a^{2}+b^{2}}.

Question 30:

Prove that the chord of the hyperbola which touches the conjugate hyperbola is bisected at the point of contact.

Cheers,

Nalin Pithwa.

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