Parabola practice · Easy

Option A (2) is correct. The length of the latus rectum is twice the perpendicular distance from the focus to the directrix.

Option B ($\frac{{125}}{{16}}$) is correct because it is the square of the distance between the point P on the parabola and the point Q where the normal at P intersects the parabola again, calculated using the parabola's equation and normal properties.

Option D (any $a>0$) is correct because the condition for collinearity of A, B, and S holds true regardless of the value of 'a'.

Option A (${p^2} + pq + 2 = 0$) is correct because it represents the relationship between the parameters p and q when the normal at one point on the parabola intersects the curve again at another point.

Option C (1/2) is correct because it is the slope that makes the tangent to the first parabola also tangent to the second.

Option A $\left( {5,2} \right)$ is correct because it represents the point of contact between the given line and the parabola, which can be found by comparing the given tangent equation with the standard form.

Option B ($x - m\mathrm{y} + a{m^2} = 0$) is correct because it represents the equation of a tangent to the parabola ${y^2} = 4ax$ when the parameter is replaced with $\frac{1}{m}$.

Option A ($a \in \left( {0,1} \right)$) is correct because it represents the range of 'a' that satisfies the condition for the point to be within the region bounded by the parabola and the line.

Option B ($\frac{{1}}{{a}}$) is correct because it represents the sum of the reciprocals of the distances from the focus to the points where mutually perpendicular tangents meet the axis.

Option A ($x=-1$) is correct because the locus of the point of intersection of perpendicular tangents to a parabola is its directrix, and for $y^2=4x$, the directrix is $x=-1$.

Option C ($a > \frac{1}{2}$) is correct because for three normals to exist from a point (a, 0) to the parabola ${y^2} = x$, the condition $a > \frac{1}{2}$ must be satisfied.

Option D ($x-2=0$) is correct because the axis of the parabola $x^2-4x-3y+10=0$ can be found by completing the square and identifying the axis of symmetry.

Option C ($x = - 1$) is correct because the locus of the point from which two perpendicular tangents are drawn to the parabola is the directrix of the parabola.

The correct answer is C ($- 4 + 3\sqrt 2$) because it is the value of $8|m_1m_2|$ calculated using the condition for common tangents to the circle and parabola.

Option D ($\sqrt{2}$) is correct because the minimum distance between the parabolas is the distance between the points where the tangents are parallel to the line of symmetry $y=x$.

Option B (G.P) is correct because SP, ST, and SQ form a geometric progression based on the properties of tangents to a parabola.

Option C ($[\pi / 2,5 \pi / 6] \cup[\pi, 3 \pi / 2]$) is correct because it represents the intervals where the point lies inside or on the parabola.

Option B ($k>1$) is correct because for three distinct normals to be drawn, the condition $k>1$ must be satisfied.

Option A (1) is correct because it represents the square of the slope of the common tangent to the parabola and the circle, derived from the tangency conditions.

Option C ((0,1)) is correct because it represents the point of contact when the straight line $x+y=1$ touches the parabola $y^2-y+x=0$.

Option C ((-4,4$\sqrt{2}$)) is correct because using the focal chord property of the parabola, the other extremity is found to be (-4,4$\sqrt{2}$).

Option A (Always in AP) is correct because the ordinates of the points A, C, and B on the parabola satisfy the condition for being in arithmetic progression.

Option C ($\pi / 3$) is correct because the angle between the tangents is found using the formula involving the slopes of the tangents and solving the quadratic equation.

Option C (${x^2} - 6x - 8y + 25 = 0$) is correct because it represents a parabola with vertex at (3,2) and focus at (3,4), derived from the given information.

Option B ($4 a \tan \alpha \sec \alpha$) is correct because it represents the length of the chord calculated using trigonometric relationships and the parabola's properties.

The correct answer is D ($9x - 4y + 4 = 0$) because it represents the tangent to the parabola that passes through the given point (4,10).

Option B (${x^2} - 4x + 8y + 12 = 0$) is correct because it represents the equation of the parabola with the given vertex and focus, derived using the definition of a parabola.

Option A ((-2,11 / 2)) is correct because by completing the square, the equation can be rewritten in standard form, revealing the vertex.

Option C ($\left( {\frac{5}{4},1} \right)$) is correct because it represents the focus of the given parabola after rewriting the equation in standard form.

Option B ($3 \sqrt{2}$) is correct because the latus rectum is twice the distance between the focus and the directrix.