Parabola practice · Mixed difficulty

Option B ((-2,0)) is correct because the locus of the intersection of perpendicular tangents is the directrix, and the point lies on both the tangent and the directrix.

Option C ($y^2=2x$) is correct because the parabola passes through the intersection points of the line and the circle.

Option D ($8\sqrt 2$) is correct because it is calculated from the distance between the latus rectum and the tangent at the vertex of the parabola.

Option B ($(y-1)^2=8(x-1)$) is correct because it represents a parabola with the given vertex and focus, opening along the positive x-axis.

The correct answer is A ($a{\left( {t + \frac{{1}}{t}} \right)^2}$) because it represents the length of the focal chord of the parabola using the parameter t.

Option C (-4) is correct because the condition for tangency is $c = -a m^2$.

Option C ($\sqrt 2$) is correct because it is the slope of the normal that subtends a right angle at the vertex of the parabola, derived by setting the sum of coefficients of x² and y² to zero.

Option D ($x-2 y+8=0$) is correct because it represents the equation of the tangent at the other end of the focal chord, calculated using the property $t_1 t_2 = -1$.

Option C (x=0) is correct because the locus of the midpoint is another parabola, and its directrix is found to be x=0.

Option A ($x + 3 = 0$) is correct because it represents the axis of the given parabola after completing the square.

Option B ((-1,6)) is correct because the focus is the midpoint of the latus rectum.

The correct answer is 05-Feb because the slope of the line through P and V is calculated by finding the point of tangency P and the vertex V of the parabola and then calculating the slope between these two points, resulting in 5/2.

Option C ($5 y^2=8 x$) is correct because the locus of the point P that divides the line segment from (0,0) to (x,y) in the ratio 2:3 is $5y^2 = 8x$.

Option D ($4{x^2} + {y^2} = 4{a^2}$) is the correct locus of point P because the chord of contact touches the rectangular hyperbola under the given conditions.

Option C (${a^2} + {b^2}$) is correct because the condition $\tan \alpha \tan \beta = 1$ leads to the relationship ${c^2} - {d^2} = {a^2} + {b^2}$.

Option B ($(k, 1) ; k>3$) is correct because completing the square gives the standard form of the parabola, and the condition for three real normals from a point on the axis leads to x > 3 and y = 1.

Option B ($y=x-1$) is correct because the equation of the chord of contact is found using the formula $T=0$.

Option B (Directrix of the parabola) is correct because the locus of the point of intersection of perpendicular tangents to a parabola is its directrix.

Option A ($x^2+4 x+8 y-4=0$) is correct because it represents a parabola with the given vertex, latus rectum length, and axis perpendicular to the original parabola.

Option A ($x^2+8y=32$) is correct because it represents a parabola with the given vertex and focus, opening downwards.

Option B ($y^2=4(a_1-a)(x-a)$) is correct because it represents a parabola with vertex at (a,0) and focus at (a1,0).

Option B ($y+2 x=0$) is correct because it represents the equation of the line passing through the vertex (0,0) and the negative end of the latus rectum.

Option D ($a{\left( {t + \frac{1}{t}} \right)^2}$) is correct because it represents the length of the focal chord of the parabola.

Option C (20) is correct because the latus rectum of a parabola is 4a, where 'a' is the distance from the vertex to the directrix.

Option B ($2 c < 2 a+d$) is correct because this condition ensures the existence of a common normal (other than the X-axis) between the two parabolas.

Option A (108) is correct because the length of the normal chord is calculated to be $6\sqrt{3}$, and squaring this value gives 108.

The correct answer is B (2) because the ratio of the areas of the two triangles is 2:1, as the area of the triangle formed by the end points of the latus rectum and a point P is twice the area of the triangle formed by the tangents at those points.

Option D (–576) is correct because after finding the focus and using the definition of a parabola, the coefficients of the resulting equation sum to -576.

Option C ($x=\frac{{B^2-A^2-C}}{{2 A}}$) is correct because it represents the x-coordinate of the latus rectum, which passes through the focus of the parabola.

Option C ($8 a$) is correct because the harmonic mean of the segments of a focal chord of a parabola is equal to its semi-latus rectum, which is $2a$ times 2, resulting in $8a$.