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.

Mathematics Hothouse