Parabola practice · Easy

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.

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 ($8 a$) 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 $8a$.

Option C ($\sqrt 2$) is correct because it is found by using the tangent condition for both the circle and the parabola and solving for |c|.

Option B ($y=\pm(x+2 a)$) is correct because by finding the tangent to the parabola and then applying the condition for tangency to the circle, we arrive at this equation.

Option B ((5,6)) is correct because it is found by using the T=S' method to find the midpoint of the chord intercepted by the line and parabola.

Option B (A parabola with vertex at $\left( {2,1} \right)$) is correct. By eliminating the parameter 't', we obtain the equation of a parabola with vertex at (2,1).

Option C ($x-y-1=0$) is correct because it represents the locus of point P derived from the condition on the inclinations of the tangents to the parabola.

Option A ($a l(l^2+2 m^2)+m^2 n=0$) is correct because it represents the condition for the line $l x+m y+n=0$ to be a normal to the parabola $y^2=4 a x$.

Option A (8) is correct because for the parabola $y^2 = 4ax$, the length of the latus rectum is $4a$, and in this case, $4a = 8$.

Option A ($\frac{a b}{b-a}$) is correct because the semi-latus rectum of a parabola is the harmonic mean of the segments of a focal chord, and solving for k gives the correct expression.

Option A ($\frac{9}{44}$) is correct because after taking the limit and simplifying the summation, the result is $\frac{9}{44}$.

The correct answer is A (${\left( {7x + 6y} \right)^2} - 570x + 750y + 2100 = 0$) because it is derived by applying the definition of a parabola: the distance to the focus equals the distance to the directrix.

Option C ((-3, -1)) is correct because the vertex of the parabola is (-3, -1) after rewriting the equation in standard form.

Option A ($y t=x+a t^2$) is correct because it represents the equation of the tangent at a point $P(t)$ to the parabola $y^2=4 a x$.

The correct answer is B (A parabola) because the parametric equations can be reduced to a quadratic equation representing a parabola.

Option B (8) is correct because substituting the given point into the parabola equation and calculating the latus rectum yields 8.

Option D ($4 y-x=36,4 y-9 x=4$) is correct because these are the equations of the tangents to the parabola that pass through the given point.

Option C (8) is correct because the length of the latus rectum is the coefficient of the y-term after completing the square.

Option D (The directrix) is correct because the locus of the poles of focal chords of a parabola is the directrix.

Option D ($2 y^2=9 a x$) is correct because the locus of the midpoint is the parabola $2y^2 = 9ax$.

Option C ($m \in ( - \infty ,0) \cup (0,\infty )$) is correct because any line passing through the focus of a parabola intersects it at two distinct points, except for the line parallel to the axis (which has a slope of 0).

Option C ($x - y + 1 = 0$) is correct because it represents a tangent to the hyperbola with a slope of 1, derived using the tangent equation formula.

Option B (($-2, 1$)) is correct because it represents the vertex of the given parabola after rewriting the equation in standard form.

Option B (A parabola) is correct because eliminating the parameter 't' results in an equation of the form $x^2 = ky$, which is a parabola.

Option C ($\frac{5}{3}$) is correct. The length of the latus rectum is found by rewriting the equation in standard form and identifying the coefficient of the y-term.

Option A ((32,32)) is correct because by using the property of focal chord $t_1t_2 = -1$ and substituting the given point, we can find the other end of the chord.

Option D (Fourth) is correct because the vertex of the parabola $(y+1)^2=-(x-1)$ is (1, -1), which lies in the fourth quadrant.

Option A ($(0,-2 a)$ and $y=2 a$) is correct because these are the focus and directrix, respectively, for the given parabola.