Parabola practice · Medium

Option A ($x + y + a = 0$) is correct because it represents the common tangent to both parabolas, derived by equating the tangent equations and solving for the condition of tangency.

Option C (4) is correct because substituting the directrix equation into the parabola equation and solving for k yields k=4 or k=-8, and 4 is one of the possible values.

Option D (none of these) is correct because the set of points from which 3 normals can be drawn is {(x, 0): x > 0}, which is not among the given options.

Option A (Directrix $4x = 3$) is correct because after finding the locus of the midpoint of the chord, the directrix of the resulting parabola is $4x=3$.

Option B (($a$, 0)) is correct because the chord joining the feet of two perpendicular normals to the parabola passes through the fixed point ($a$, 0).

Option A ($\ell n=a m^2$) is correct because this condition ensures that the line is tangent to the parabola, derived from the tangency condition c = a/m.

Option C ($m^3 y=2 a m^2-m^2 x+a$) is correct because it represents the equation of the normal to the parabola at the given point after simplification.

Option D ($90^{\circ}$) is correct because the point (-2a, 2) lies on the directrix of the parabola, implying the tangents from this point are perpendicular.

Option B (8) is correct because the focal distance of the point on the parabola is 8.

Option D ($y=x+2$) is correct because it represents a common tangent to both the parabola and the hyperbola, derived from the tangency conditions.

The correct answer is D (24) because substituting the normal form into the parabola equation and solving for k yields 24.

The correct answer is A (4) because the minimum value of c is found by using the tangent equation and applying AM-GM inequality.

Option A (Always in AP) is correct because the ordinate of the intersection point C is the average of the ordinates of A and B, indicating an arithmetic progression.

Option A is correct because the locus of the intersection point of the perpendicular tangents to the given parabolas is the line $x + 3 = 0$.

Option B ((2,2)) is correct because it is the point that satisfies both parametric equations when eliminating the parameters t and s.

Option C ((0,1)) is correct because it represents the set of positive values of m for which a line with slope m is a common tangent to the given ellipse and parabola.

Option D (-3) is correct because it uses the formula relating the parameters of the points where the normal intersects the parabola.

Option C (HP) is correct because if $t_1, t_2, t_3$ are in AP, then $t'_1, t'_2, t'_3$ are in HP, where $t'_i = -1/t_i$.

Option A (3) is correct because the angle between the tangents is found using the locus of the point of intersection of tangents and the given parabola equation.

Option B (9) is correct because substituting the normal form into the given equation yields k = 9.

Option C ($c=-a m^2$) is correct because it is the condition for the line to be tangent to the parabola.

Option A $\left(x_1, y_1\right)$ is correct because the midpoint of the chord QR is the same as the point P on the original parabola.

Option B ($x + a = 0$) is correct because the locus of the intersection of perpendicular tangents to a parabola is its directrix.

The correct answer is ${x^2} - {y^2} - 6ax + 9{a^2} = 0$ because it represents the combined equation of the normals at the ends of the latus rectum of the given parabola.

Option A (8) is correct because after finding the incenter and the 'a' value of the parabola, we calculate the focus and then compute the required ratio.

Option C (${a^2}{l^2} + {b^2}{m^2} = 2{n^2}$) is correct because it is derived by comparing the given line with the equation of the chord joining two points on the ellipse with eccentric angles differing by 90 degrees.

Option C ($y^2-2 a x+2 a^2=0$) is correct because it is the locus of the midpoints of the chords of the parabola that pass through the focus, found by using the equation T = S' and the condition that the chord passes through the focus.

Option D ($32 \sqrt{2}$) is correct because the length of the chord through L with slope -1 is calculated to be $32 \sqrt{2}$.

Option A (3) is correct because the tangent of the angle between the two tangents is calculated using the slopes of the tangents and the formula for the angle between two lines.

Option C (a pair of straight lines) is correct because the discriminant $\Delta = 0$ and $h^2 = ab$, indicating a pair of parallel straight lines.