Find the equations of tangents and normal for the circle: x² + y² = 5 at (1,2).

Find the equations of tangent and normal for the circle: x² + y² = 25 at (3,4).

Find the equation of tangent and normal to the circle: x² + y² = 100 at (1,2).

Find the equation of tangent and normal to the circle: x² + y² = 169 at (12,-5).

Find the equation of tangent and normal to circle: x² + y² - 2x - 4y + 3 = 0 at (2,3).

Find the equation of tangent and normal to the circle: x² + y² - 3x + 4y - 31 = 0 at (-2,3).

Find the length of the intercept made by the straight line x + y = 3 with the circle x² + y² - 2x - 3 = 0.

Find the length of the intercepts made by the straight line 2x - y = 7 with the circle x² + y² - 6x - 8y + 15 = 0.

Write the condition for a line y = mx + c to be tangent to the circle x² + y² = a².

Show that the line 3x + 4y - 20 = 0 touches the circle x² + y² = 16. Also, find the point of contact.

Show that the line 3x - 4y = 25 and the circle x² + y² = 25 intersect at a coincident point.

Define tangent and normal to a circle.

Find the equation of the tangent to the circle x² + y² = 25 inclined at an angle of 60^o to the x-axis.

Find the equation of the tangent to the circle x² + y² = 9 parallel to 3x + 4y = 0.

Find the equation of the tangent to the circle x² + y² -6x + 4y = 12 and parallel to the line 4x + 3y + 5 = 0.

Find the equation of the tangents to the circle x² + y² = 5, which are perpendicular to the line x + 2y = 0.

Find the equation of the tangents to the circle x² + y² - 2x - 4y -4 = 0 which are perpendicular to the line 3x - 4y = 1.

Find the equation of the tangent to the circle x² + y² = 10 at the point whose abscissa is 1.

Find the equation of the tangent to the circle x + y - 2ax = 0 at $\rm (a (1 + \cos \alpha ), a \sin \alpha)$.

Find the value of k if the line 2x - y + k = 0 may touch the circle x² + y² = 5.

Find the value of k if the line 4x - 3y + k = 0 is tangent to the circle x² + y² - 8x + 12y + 3 = 0.

Find the value of k if the line $\rm x \cos \alpha + y \sin \alpha = k$ is a tangent to the circle x² + y² - 2ax $\rm\cos \alpha - 2ay \sin \alpha$ = 0.

Prove that the tangents to the circle x² + y² = 169 at (5,12) and (-5,-12) are parallel.

Prove that the tangents to the circle x² + y² = 25 at (3,4) and (4,-3) are perpendicular.

Find the condition that the line lx + my + n = 0 may be a tangent to the circle: x² + y² = a².

Find the condition that the line lx + my + n = 0 may be a tangent to the circle: x² + y² + 2gx + 2fy + c = 0.

Find the condition that the line lx + my + n = 0 may be a normal to the circle x² + y² + 2gx + 2fy + c = 0.

Find the condition that the line $\rm \frac{x}{a} + \frac{y}{b} = 1$ may touch the circle: $\rm (x - a)^2 + (y - b)^2 = r^2$.

Write the conditions for two circles to touch each other. Find the condition for the two circles x² + y² = a² and (x - c)² + y² = b² to touch (i) externally and (ii) internally.

Find the equation of the circle whose center is at (h,k) and which passes through the origin. Prove that the equation of the tangent at the origin is hx + ky = 0.

Find the equation of tangents drawn from an external point (13,0) to the circle x² + y² = 25.

Find the equation of tangents drawn from an external point (4,-2) to the circle x² + y² = 10.

Find the equation of tangents drawn from an external point (5,-1) to the circle x² + y² = 8.

Find the equation of tangents drawn from an external point (0,1) to the circle x² + y² - 2x + 4y = 0.

Find the length of the tangent to the circle x² + y² = 9 from (3,2).

Find the length of the tangent to the circle 2x² + 2y² = 5 from (4,-1).

Find the length of the tangent to the circle x² + y² - 2x - 3y -1 = 0 from (2,5).

Find the length of the tangent to the circle x² + y² - 4y - 15 = 0 from (4,5).

Find the value of k so that the length of the tangent from (5,4) to the circle x² + y² + 2ky = 0 is 1.

Find the value of k so that the length of the tangent from (4,5) to the circle x² + y² - ky - 5 = 0 is 5.

Prove that the straight line y = x + a$\rm \sqrt{2}$ touches the circle x² + y² = a². Also, find the point of contact.

Find the equation of the line through the point (1, -1) which cuts off a chord of length 4$\rm \sqrt{3}$ from the circle x² + y² - 6x - 4y - 3 = 0.

Find the equation of a parabola whose vertex is at (0,0) and focus at (-3/2,0). Also, find its focal length and where the parabola is opened.

Find the equation of a parabola whose vertex is at (0,0) and focus at (3,0). Also, find its focal length and where the parabola is opened.

Find the equation of a parabola whose vertex is at (0,0) and focus at (0,4). Also, find its focal length and where the parabola is opened.

Find the equation of a parabola whose vertex is at (3,2) and focus at (6,2). Also, find its focal length and where the parabola is opened.

Find the equation of a parabola whose vertex is at (5,3) and focus at (5,6). Also, find its focal length and where the parabola is opened.

Find the equation of the parabola whose focus is at (4,0) and its directrix is x = -4.

Find the equation of the parabola whose vertex is at the origin, passes through the point (5,2), and symmetric about the y-axis.

Find the equation of a parabola whose length of latus rectum is 16, axis parallel to the x-axis, and passing through (3,2) and (3,-2).

For the parabola 3y² = 8x, find:

1. the coordinates of the vertex
2. the coordinates of the focus
3. equation axis of the parabola
4. equation of directrix
5. length of latus rectum
6. the coordinates of the ends of the latus rectum

For the parabola x² = -8y, find:

1. the coordinates of the vertex
2. the coordinates of the focus
3. equation axis of the parabola
4. equation of directrix
5. length of latus rectum
6. the coordinates of the ends of the latus rectum

For the parabola (y - 2)² = 2(x + 3), find:

1. the coordinates of the vertex
2. the coordinates of the focus
3. equation axis of the parabola
4. equation of directrix
5. length of latus rectum
6. the coordinates of the ends of the latus rectum

For the parabola x² - 4x - 3y + 13 = 0 find:

1. the coordinates of the vertex
2. the coordinates of the focus
3. equation axis of the parabola
4. equation of directrix
5. length of latus rectum
6. the coordinates of the ends of the latus rectum

Find the focal distance of a point P(2,4) for the parabola y² = 8x.

Find the focal distance of a point P(4,1) for the parabola x² = 16y.

At what points on the parabola y² = 36x is the ordinate two times that of its abscissa?

Find the area of the triangle formed by the lines joining the vertex of the parabola x² = 12y to the ends of its latus rectum.

Find the locus of the mid-points of the chords of a parabola y² = 4ax, drawn from focus. Prove that the locus is a parabola.

A double ordinate of the curve y² = 4ax is of length 8a. Prove that the lines joining the vertex to its ends are at right angles.

Find the parametric equation of the parabola y² = 10x.

Find the parametric equation of the parabola x² = -16y.

Find the parametric equation of the parabola (y - 2)² = 2(x + 3).

The equation of tangent (3, 4) to the circle x² + y² = 25 is

Find the equations of tangents and normal for the circle: x² + y² = 5 at (1,2).

Find the equations of tangent and normal for the circle: x² + y² = 25 at (3,4).

Find the equation of tangent and normal to the circle: x² + y² = 100 at (1,2).

Find the equation of tangent and normal to the circle: x² + y² = 169 at (12,-5).

Find the equation of tangent and normal to circle: x² + y² - 2x - 4y + 3 = 0 at (2,3).

Find the equation of tangent and normal to the circle: x² + y² - 3x + 4y - 31 = 0 at (-2,3).

Find the length of the intercept made by the straight line x + y = 3 with the circle x² + y² - 2x - 3 = 0.

Find the length of the intercepts made by the straight line 2x - y = 7 with the circle x² + y² - 6x - 8y + 15 = 0.

Write the condition for a line y = mx + c to be tangent to the circle x² + y² = a².

Show that the line 3x + 4y - 20 = 0 touches the circle x² + y² = 16. Also, find the point of contact.

Show that the line 3x - 4y = 25 and the circle x² + y² = 25 intersect at a coincident point.

Define tangent and normal to a circle.

Find the equation of the tangent to the circle x² + y² = 25 inclined at an angle of 60^o to the x-axis.

Find the equation of the tangent to the circle x² + y² = 9 parallel to 3x + 4y = 0.

Find the equation of the tangent to the circle x² + y² -6x + 4y = 12 and parallel to the line 4x + 3y + 5 = 0.

Find the equation of the tangents to the circle x² + y² = 5, which are perpendicular to the line x + 2y = 0.

Find the equation of the tangents to the circle x² + y² - 2x - 4y -4 = 0 which are perpendicular to the line 3x - 4y = 1.

Find the equation of the tangent to the circle x² + y² = 10 at the point whose abscissa is 1.

Find the equation of the tangent to the circle x + y - 2ax = 0 at $\rm (a (1 + \cos \alpha ), a \sin \alpha)$.

Find the value of k if the line 2x - y + k = 0 may touch the circle x² + y² = 5.

Find the value of k if the line 4x - 3y + k = 0 is tangent to the circle x² + y² - 8x + 12y + 3 = 0.

Find the value of k if the line $\rm x \cos \alpha + y \sin \alpha = k$ is a tangent to the circle x² + y² - 2ax $\rm\cos \alpha - 2ay \sin \alpha$ = 0.

Prove that the tangents to the circle x² + y² = 169 at (5,12) and (-5,-12) are parallel.

Prove that the tangents to the circle x² + y² = 25 at (3,4) and (4,-3) are perpendicular.

Find the condition that the line lx + my + n = 0 may be a tangent to the circle: x² + y² = a².

Find the condition that the line lx + my + n = 0 may be a tangent to the circle: x² + y² + 2gx + 2fy + c = 0.

Find the condition that the line lx + my + n = 0 may be a normal to the circle x² + y² + 2gx + 2fy + c = 0.

Find the condition that the line $\rm \frac{x}{a} + \frac{y}{b} = 1$ may touch the circle: $\rm (x - a)^2 + (y - b)^2 = r^2$.

Write the conditions for two circles to touch each other. Find the condition for the two circles x² + y² = a² and (x - c)² + y² = b² to touch (i) externally and (ii) internally.

Find the equation of the circle whose center is at (h,k) and which passes through the origin. Prove that the equation of the tangent at the origin is hx + ky = 0.

Find the equation of tangents drawn from an external point (13,0) to the circle x² + y² = 25.

Find the equation of tangents drawn from an external point (4,-2) to the circle x² + y² = 10.

Find the equation of tangents drawn from an external point (5,-1) to the circle x² + y² = 8.

Find the equation of tangents drawn from an external point (0,1) to the circle x² + y² - 2x + 4y = 0.

Find the length of the tangent to the circle x² + y² = 9 from (3,2).

Find the length of the tangent to the circle 2x² + 2y² = 5 from (4,-1).

Find the length of the tangent to the circle x² + y² - 2x - 3y -1 = 0 from (2,5).

Find the length of the tangent to the circle x² + y² - 4y - 15 = 0 from (4,5).

Find the value of k so that the length of the tangent from (5,4) to the circle x² + y² + 2ky = 0 is 1.

Find the value of k so that the length of the tangent from (4,5) to the circle x² + y² - ky - 5 = 0 is 5.

Prove that the straight line y = x + a$\rm \sqrt{2}$ touches the circle x² + y² = a². Also, find the point of contact.

Find the equation of the line through the point (1, -1) which cuts off a chord of length 4$\rm \sqrt{3}$ from the circle x² + y² - 6x - 4y - 3 = 0.

Find the equation of a parabola whose vertex is at (0,0) and focus at (-3/2,0). Also, find its focal length and where the parabola is opened.

Find the equation of a parabola whose vertex is at (0,0) and focus at (3,0). Also, find its focal length and where the parabola is opened.

Find the equation of a parabola whose vertex is at (0,0) and focus at (0,4). Also, find its focal length and where the parabola is opened.

Find the equation of a parabola whose vertex is at (3,2) and focus at (6,2). Also, find its focal length and where the parabola is opened.

Find the equation of a parabola whose vertex is at (5,3) and focus at (5,6). Also, find its focal length and where the parabola is opened.

Find the equation of the parabola whose focus is at (4,0) and its directrix is x = -4.

Find the equation of the parabola whose vertex is at the origin, passes through the point (5,2), and symmetric about the y-axis.

Find the equation of a parabola whose length of latus rectum is 16, axis parallel to the x-axis, and passing through (3,2) and (3,-2).

For the parabola 3y² = 8x, find:

1. the coordinates of the vertex
2. the coordinates of the focus
3. equation axis of the parabola
4. equation of directrix
5. length of latus rectum
6. the coordinates of the ends of the latus rectum

For the parabola x² = -8y, find:

1. the coordinates of the vertex
2. the coordinates of the focus
3. equation axis of the parabola
4. equation of directrix
5. length of latus rectum
6. the coordinates of the ends of the latus rectum

For the parabola (y - 2)² = 2(x + 3), find:

1. the coordinates of the vertex
2. the coordinates of the focus
3. equation axis of the parabola
4. equation of directrix
5. length of latus rectum
6. the coordinates of the ends of the latus rectum

For the parabola x² - 4x - 3y + 13 = 0 find:

1. the coordinates of the vertex
2. the coordinates of the focus
3. equation axis of the parabola
4. equation of directrix
5. length of latus rectum
6. the coordinates of the ends of the latus rectum

Find the focal distance of a point P(2,4) for the parabola y² = 8x.

Find the focal distance of a point P(4,1) for the parabola x² = 16y.

At what points on the parabola y² = 36x is the ordinate two times that of its abscissa?

Find the area of the triangle formed by the lines joining the vertex of the parabola x² = 12y to the ends of its latus rectum.

Find the locus of the mid-points of the chords of a parabola y² = 4ax, drawn from focus. Prove that the locus is a parabola.

A double ordinate of the curve y² = 4ax is of length 8a. Prove that the lines joining the vertex to its ends are at right angles.

Find the parametric equation of the parabola y² = 10x.

Find the parametric equation of the parabola x² = -16y.

Find the parametric equation of the parabola (y - 2)² = 2(x + 3).

The equation of tangent (3, 4) to the circle x² + y² = 25 is